eigenvalue processes of and their overlaps july 2020

Eigenvalue processes of

Elliptic Ginibre Ensembleand their Overlaps

July 2020

SATOSHI YABUOKU

Graduate School of

Science and Engineering

CHIBA UNIVERSITY

(千葉大学審査学位論文)Eigenvalue processes of

Elliptic Ginibre Ensembleand their Overlaps

July 2020

SATOSHI YABUOKU

Graduate School of

Science and Engineering

CHIBA UNIVERSITY

Contents

1 Random Matrix Theory 61.1 Gaussian Ensembles . . . . . . . . . . . . . . . . . . . . . . . 61.2 Ginibre Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Elliptic Ginibre Ensemble . . . . . . . . . . . . . . . . . . . . 171.4 Eigenvector correlations . . . . . . . . . . . . . . . . . . . . . 19

2 Matrix-valued processes 212.1 Dyson’s Brownian motions . . . . . . . . . . . . . . . . . . . . 212.2 Some normal matrix-valued processes . . . . . . . . . . . . . . 23

2.2.1 Unitary matrix-valued processes . . . . . . . . . . . . . 232.2.2 Wishart processes . . . . . . . . . . . . . . . . . . . . . 23

2.3 Time-depending Ginibre Ensemble and Overlaps . . . . . . . . 26

3 Matrix-valued process associated with Elliptic Ginibre Ensem-ble 323.1 Settings and Main Results . . . . . . . . . . . . . . . . . . . . 333.2 Proofs of main results . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . 373.2.2 Proof of Lemma 3.2.1 . . . . . . . . . . . . . . . . . . . 433.2.3 Proof of Corollary 3.1.6 . . . . . . . . . . . . . . . . . 49

A Tools and basic properties 54

1

Preface

The study of random matrices has begun for multivariate analysis in 1928by Wishart in [73]. A breakthrough of Random Matrix Theory (RMT) camein the 1950s by Wigner in [69–72]. He considered Hermitian random matri-ces, whose entries are independent Gaussians, to evaluate nuclear energy levelsin quantum mechanics. These random matrix models are called Gaussian Or-thogonal, Unitary and Symplectic ensemble, and the study of random matriceswas developed by Dyson, Gaudin, and Mehta in [17,18,20,21,27,44,45,47–50].We denote the three models by GOE, GUE and GSE for short, respectively.The joint density function of N eigenvalues has the following form:

p(x1, . . . , xN) ∝ exp

(−β

4

N∑i=1

x2i

) ∏1≤i<j≤N

|xi − xj|β,

where β = 1, 2, 4 correspond to GOE, GUE and GSE, respectively. In the casesof the three ensembles, as N → ∞, the empirical spectral distribution (ESD)1N

∑Ni=1 δ λi√

N

of the eigenvalues λi, i = 1, . . . , N converges to a deterministic

distribution, which has density

p(x) =1

2π

√4 − x2,

on the interval [-2,2]. This phenomenon is called Wigner’s semi-circle law, andthe universality is also known; the convergence of ESDs does not depend onthe distributions of matrix entries in [54,71].

In particular, GUE has useful determinantal structures to analyze the be-havior of eigenvalues, so that the spacing distribution and fluctuations of thelargest eigenvalues with proper scaling were also obtained. These results arealso universal, showed later by Tao and Vu in [61].

In 1965, Ginibre suggested the modifications of GOE, GUE and GSEto non-symmetric matrices and considered the complex eigenvalues with hismathematical interests in [28]. He found that N complex eigenvalues of non-symmetric matrices, whose entries are independent complex Gaussians, has

2

PREFACE 3

density

p(z1, . . . , zN) ∝ exp

(−

N∑i=1

|zi|2) ∏

1≤i<j≤N

|zi − zj|2.

This model is called Ginibre ensemble (GE). Mehta showed in [46] that theESD of GE converges to the uniform distribution on the unit disc in C. Thisphenomenon, known as the circular law, was expected to be universal, andwith contributions by Girko, Edelman, Bai, Gotze and Tikhomirov and othersin [5, 6, 9, 23,29,31,32], finally Tao and Vu proved the conjecture in [60].

The two models of GUE and GE have completely different matrix structure;the former is Hermitian, and the latter is non-symmetric. In the 1980s, Girkoin [30] suggested the interpolation model of the two random matrices, andSommers, Crisanti, Sompolinsky and Stein parametrized his models by τ ∈[−1, 1] in [58]. The parameter τ implies the matrix Hermiticity. With τ = 1,the matrix is Hermitian and thus GUE, and with τ = 0, it is completely non-Hermitian and thus GE. This matrix model is called Elliptic Ginibre Ensemble(EGE) and they found that for −1 < τ < 1, the joint density function of N

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

re_j

im_j

Figure 1. Plots of eigenvalues of EGE scaled by N− 12 , N = 3000, τ = 0.3.

complex eigenvalues is

p(z1, . . . , zN) ∝ exp[− 1

1 − τ 2(

N∑i=1

|zi|2 −τ

2

N∑i=1

(z2i + z∗i2))]∏

i<j

|zi − zj|2,

4 PREFACE

and Girko showed that the ESD converges to the uniform distribution of theellipse parametrized by τ . Taking τ = 0, we obtain the circular law. Thisphenomenon is called Girko’s elliptic law, and the universality also holds in[51].

We now discuss time evolution models of random matrices. In 1962, Dysonconsidered the time evolutions of GOE, GUE and GSE in [19]. He found outthe log-gas analogy in the forms of the joint density of the three models, whichis described as the first formula in the preface, and derived the stochasticdifferential equations (SDEs) of the real eigenvalues:

dλi(t) = dBi(t) +β

2

∑j(=i)

1

λi(t) − λj(t)dt, i = 1, . . . , N,

where Bi, i = 1, . . . , N , are independent one-dimensional Brownian motionsand β = 1, 2, 4 correspond to GOE, GUE and GSE, respectively. This modelis called Dyson’s Brownian motion models. We remark that the eigenvalueprocesses are diffusions, and these processes inherit good properties of thestatic cases. He also stated that processes satisfying the above SDEs agreewith non-colliding Brownian motions, and later this claim was justified byusing stochastic analysis in [38–40].

Recently, the time evolution model of GE has been studied. This mod-ification is very natural in RMT, as mentioned above in the static models.Nevertheless, this model is far from Dyson’s models. The main reason is thatin non-normal matrix cases, eigenvalues are very sensitive for perturbations ofmatrices, and so eigenvectors effect the behavior of dynamics of eigenvalues.The quantification of the sensitivity is given by so called overlaps Oij, whichare defined by inner products of right and left eigenvectors and explain thenon-orthogonality of eigenvectors. In the 1990s, Chalker and Mehlig analyzedoverlaps of static GE in [14, 15] and obtained the asymptotic behavior of theconditional expectation of the diagonal overlap as

E[O11|λ1 = z] ∼ N(1 − |z|2), N → ∞.

In 2018, Bourgade and Dubach showed in [10] that for static GE, the limitingconditional distribution of the diagonal overlap converges to inverse Gammadistribution with parameter 2:

O11

N(1 − |z|2)(d)→ 1

γ2, conditioned on λ1 = z.

They also mentioned that the time evolution model of GE is characterized bytime-depending overlaps Oij(t), without using their explicit forms.

On the basis of the above results and observations, we consider a time evo-lution model of non-symmetric matrices, in particular, the dynamics of EGE.

PREFACE 5

This model includes Dyson’s model of GUE and the time evolution of GEby using the Hermiticity parameter τ . Using a different approach of previousstudies, we obtain the SDEs explicitly which eigenvalue processes of EGE sat-isfy in [74]. By virtue of the parametrization τ , we can see the deformations ofthe eigenvalues and time-depending overlaps Oij(t) with respect to the matrixnormality.

The organization of this thesis is the following. In Chapter 1, we intro-duce classical random matrix models, GOE, GUE, GSE, GE and EGE, andtheir classical results. Overlaps are also mentioned. In Chapter 2, we referto matrix-valued processes. Dyson’s models and Wishart processes are repre-sented as non-colliding diffusion systems. For the time evolution of GE, thesolution of the Fokker-Planck equation of eigenvalues and eigenvectors is ob-tained. In Chapter 3, we discuss the time evolution of EGE and show ourmain results and proofs. In Appendix, we put together some tools, which aremainly used in Section 3.2.

Chapter 1

Random Matrix Theory

In this Chapter, we introduce random matrix models. Firstly, we summarizewell known results of three Gaussian ensembles. Secondly, Ginibre ensembleand the circular law are discussed. Thirdly, Elliptic Ginibre ensemble and theHermiticity of matrices are discussed. Finally, we introduce eigenvector corre-lations and refer to recent results showed by Bourgade and Dubach. Each of allmatrix ensembles introduced in this Chapter are discussed as time evolutionmodels in Chapters 2 and 3.

1.1 Gaussian Ensembles

We consider an N × N symmetric matrix H(1)N = h(1)ij 1≤i,j≤N whose entries

are given by independent Gaussians:

h(1)ij :=

√2gii, i = j,

gij, i < j,

where gij has density function 1√2πe−

x2

2 . Then H(1)N is called Gaussian Or-

thogonal Ensemble (GOE). Similarly, we define Hermitian matrices H(2)N

and H(4)N by

h(2)ij :=

gii, i = j,

g(1)ij +

√−1g

(2)ij√

2, i < j,

h(4)ij :=

gii√

2, i = j,

g(1)ij + ig

(2)ij + jg

(3)ij + kg

(4)ij

2, i < j,

6

1.1. GAUSSIAN ENSEMBLES 7

where g(k)ij are independent standard Gaussians and i, j, k are quaternions.

Then H(2)N (respectively H

(4)N ) is called Gaussian Unitary Ensemble (GUE)

(respectively Gaussian Symplectic Ensemble (GSE)). We denote H(β)N ,

β = 1, 2, 4 as the above three matrix models. H(1)N is distributed on the N ×N

symmetric matrix space SN and H(2)N is distributed on the N ×N Hermitian

matrix space HN . We consider the bijection map SN → RN(N+1)

2 defined bytaking on-or-above-diagonal entries as coordinates and normalize Lebesgue

measure on SN to push forward to Lebesgue measure on RN(N+1)

2 . Then, bythe definition of GOE, the density function of H

(1)N with respect to Lebesgue

measure on SN is

p(H) =1

c(1)N

exp

(−1

4

N∑i=1

H2ii −

1

2

∑1≤i<j≤N

H2ij

)

=1

c(1)N

exp

(−1

4tr(H2)

), (1.1)

where c(1)N = 2

N2 (2π)

N(N+1)4 . Similarly, the density function of H

(2)N with respect

to Lebesgue measure on HN is

p(H) =1

c(2)N

exp

(−1

2tr(H2)

), (1.2)

where c(2)N = 2

N2 π

N2

2 . By the property of trace, the distributions of GOE as(1.1) and GUE as (1.2) are invariant under the transform H → U∗HU , whereU is an N × N orthogonal matrix for GOE or unitary matrix for GUE. Inparticular, let λ1 ≤ · · · ≤ λN be the real eigenvalues of GOE or GUE, then

tr(H2) = tr(U∗HUU∗HU) = tr(D2) =N∑i=1

λ2i ,

where D = diag(λ1, . . . , λN). Hence the density functions of GOE and GUEonly depend on their eigenvalues. From now on, we focus on the distributionsof eigenvalues of H

(β)N , β = 1, 2, 4. A classical and important result is that the

joint probability distribution functions (jpdf for short) of the eigenvalues aredescribed explicitly as following.

Theorem 1.1.1 ( [4]). For β = 1, 2, 4, the jpdf of the ordered eigenvalues of

H(β)N with respect to Lebesgue measure on RN is

pN(x1, . . . , xN) =1

C(β)N

1x1≤···≤xNexp

(−β

4

N∑i=1

x2i

) ∏1≤i<j≤N

|xi − xj|β, (1.3)

8 CHAPTER 1. RANDOM MATRIX THEORY

where

C(β)N = (2π)

N2

(β

2

)−βN(N−1)

4−N

2N∏j=1

Γ(jβ2

)Γ(β2

) ,Γ(s) :=

∫ ∞

0

xs−1e−xdx, s > 0.

Remark 1.1.2. In the equation (1.3), the difference-product terms∏

1≤i<j≤N |xi−xj| can be written as the modulus of Vandermonde determinant:

∆(xxx) :=N

deti,j=1

xj−1i = det

1 1 · · · 1x1 x2 · · · xNx21 x22 · · · x2N...

......

...xN−11 xN−1

2 · · · xN−1N

=∏

1≤i<j≤N

(xj − xi).

For β = 1, 2, |∆(xxx)|β is obtained by the Jacobian of the diagonalizing trans-form H = UDU∗. In general, Theorem 1.1.1 is derived by Weyl’s integrationformula, see [4, section 4.1].

Next, we consider general random Hermitian matrices called Wigner ma-trices, which include the above three models.

Definition 1.1.3 (Wigner matrix). Wigner matrix WN = ξij1≤i,j≤N isdefined as a Hermitian matrix whose entries are independent and identicallydistributed (i.i.d. for short) and satisfy the following second moment condi-tions:

E[ξij] = 0, E[ξ2ii] = E[|ξij|2] = 1.

λ1(WN) ≤ · · · ≤ λN(WN) denote the eigenvalues of WN , and define theempirical spectral distribution (ESD for short)

µ 1√NWN

:=1

N

N∑i=1

δ 1√Nλi(WN ),

where for any A ∈ B(R),

δx(A) :=

1, x ∈ A,

0, x /∈ A.

µ 1√NWN

is a random measure on R. Recall that a sequence of probability

measures µn∞n=1 on R converges to µ if and only if for any bounded continuousfunction f on R,

< µn, f > → < µ, f >,


where < µ, f >:=∫f(x)dµ(x). Then for µ 1√

NWN

, the Wigner’s semi-circle

law holds.

Theorem 1.1.4 (Semi-circle law). Let WN be a Wigner matrix, then

µ 1√NWN

→ µsc, N → ∞, a.s.,

where µsc has density 12π

√4 − x2 on the interval [-2,2].

Histogram of ev_GUE

ev_GUE

Density

-3 -2 -1 0 1 2 3

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Figure 1.1. Histgram of eigenvalues of WN with independent Gaussian entries scaled byN− 1

2 , N = 3000.

One of the ways to show the semi-circle law is the moment method andconcentrate inequalities in [4,59]. Another powerful tool is the Stieltjes trans-formation: for z ∈ C \ R,

sµ 1√N

WN(z) :=

∫R

1

x− zdµ 1√

NWN

(x) =1

Ntr

((1√NWN − zIN

)−1),

in [6, 54,59].


Turning to Gaussian ensembles, we analyze the spacings of the eigenvalues.In particular, GUE has useful determinantal structures, and we deal withthem. Similar deformations hold for GOE and GSE. By the equation (1.3),the N unordered eigenvalues have jpdf

pN(x1, . . . , xN) =1

N !C(2)N

exp

(−1

2

N∑i=1

x2i

) ∏1≤i<j≤N

|xi − xj|2. (1.4)

We define the k-correlation functions as

RN,k(x1, . . . , xk) :=N !

(N − k)!

∫p(x1, . . . , xN)

N∏i=k+1

dxi, 1 ≤ k ≤ N.

To analyze spacing distributions, we rewrite the jpdf (1.4) by using Hermitepolynomials.

Definition 1.1.5 (Hermite polynomials, oscillator-wave functions).The n-th Hermite polynomial Hn(x) is defined by

Hn(x) := (−1)nex2

2dn

dxne−

x2

2 , n ∈ 0 ∪ N,

and the n-th normalized oscillator wave-function ψn(x) is defined by

ψn(x) :=e−

x2

4 Hn(x)

(√

2πn!)12

.

Note that the oscillator wave-functions have the orthogonal property:∫ψk(x)ψℓ(x)dx = δkℓ. (1.5)

By the monic property of the Hermite polynomials, the Vandermonde deter-minant factors in (1.4) is described as

∆(xxx) =N

deti,j=1

xj−1i =

N

deti,j=1

Hj−1(xi). (1.6)

Using (1.5) and (1.6) and applying Andreief’s identity, we have the following.

Proposition 1.1.6. [48, (5.2.14)] For any 1 ≤ k ≤ N ,

RN,k(x1, . . . , xk) =k

deti,j=1

K(N)(xi, xj),


where the kernel K(N)(x, y) is

K(N)(x, y) :=N−1∑k=0

ψk(x)ψk(y) =√NψN(x)ψN−1(y) − ψN−1(x)ψN(y)

x− y, (1.7)

and K(N)(x, y) satisfies∫K(N)(x, y)K(N)(y, z)dy = K(N)(x, z).

Here, the second equation in (1.7) is called Christoffel-Darboux for-mula. In particular, we take k = N in the Proposition 1.1.6 and obtain

pN(x1, . . . , xN) =1

N !

N

detk,ℓ=1

K(N)(xk, xℓ). (1.8)

Remark that using the determinantal form of (1.8) and kernels obtained in therest of this Subsection 1.1, we can construct a determinantal point process(DPP for short), see for example, [4, Section 4.2] and [35]. Consequently, theprobability for the ordered eigenvalues of GUE λ1 ≤ · · · ≤ λN has the followingexpression.

Proposition 1.1.7. ( [4, Lemma 3.2.4]) For any A ∈ B(R),

P

(N∩i=1

λi ∈ A

)= 1 +

∞∑k=1

(−1)k

k!

∫Ac

· · ·∫Ac

k

deti,j=1

K(N)(xi, xj)k∏

i=1

dxi. (1.9)

Note that the summation in (1.9) is actually finite. This expression is aFredholm determinant, which is a useful tool for the analysis of spacingdistributions in [64, 65]. Indeed, if we show a convergence of a kernel, thenwe obtain the convergence of the Fredholm determinant with respect to thekernel. By using Proposition 1.1.7 and the asymptotic estimates of K(N)(x, y)with proper scaling, we have the following result on the spacing distributionin the bulk.

Theorem 1.1.8. (Gaudin-Mehta, [4, Theorem 3.1.1]) For any compact setA ⊂ R,

limN→∞

P

(N∩i=1

√Nλi /∈ A

)= 1 +

∞∑k=1

(−1)k

k!

∫A

· · ·∫A

k

deti,j=1

Ksine(xi, xj)k∏

i=1

dxi,

where Ksine is the Sine kernel:

Ksine(x, y) :=

1

π

sin(x− y)

x− y, x = y,

1

π, x = y.


In Theorem 1.1.8, this scaling is called bulk scaling. Roughly speaking,the semi-circle law (Theorem 1.1.4) implies that the maximum eigenvalue λN ofGUE is distributed around 2

√N . Indeed, we can show λN√

N→ 2, in probability

[4, Theorem 2.1.22]. The next result states that under a proper scaling, thefluctuations of λN√

N−2 appear and are described explicitly. We define the Airy

function as

Ai(x) :=1

2πi

∫C

eζ3

3−xζdζ,

where C is the contour consisting of the ray joining e−πi3 ∞ to the origin plus

the ray joining the origin to eπi3 ∞. Then, for the spacing distribution in the

edge, we have the following:

Theorem 1.1.9. (Tracy-Widom [64], [4, Theorem 3.1.4 and Theorem 3.1.5])

limN→∞

P

(N

23

(λN√N

− 2

)≤ t

)= 1 +

∞∑k=1

(−1)k

k!

∫ ∞

t

· · ·∫ ∞

t

k

deti,j=1

A(xi, xj)k∏

i=1

dxi

:= F2(t),

where A(x, y) is the Airy kernel:

A(x, y) :=Ai(x)Ai′(y) − Ai′(x)Ai(y)

x− y.

Moreover, F2(·) is a distribution function and has the representation

F2(t) = exp

(−∫ ∞

t

(x− t)q(x)2dx

),

where q satisfies the Painleve II equation:

q′′ = tq + 2q3,

and q(t) ∼ Ai(t), as t→ ∞.

In Theorem 1.1.9, this scaling is called soft-edge scaling. At the end ofthe subsection, we remark that the above results of GUE are also known forGOE and GSE in [4,66,67]. Furthermore, the above three Gaussian ensemblesare generalized; we can construct random matrices whose eigenvalues have thejpdf

pN(x1, . . . , xN) ∝ exp

(−β

4

N∑i=1

x2i

) ∏1≤i<j≤N

|xi − xj|β, β > 0,

by tridiagonal matrix forms [22]. This model is called Gaussian beta en-sembles (GβE). As an application of GβE, for β > 0, the Tracy-Widom lawis studied in [55].

1.2. GINIBRE ENSEMBLE 13

1.2 Ginibre Ensemble

Ginibre extended an analogy of Gaussian ensembles, GOE, GUE and GSE, tothe non-symmetric matrices, which have complex eigenvalues. In the rest ofthis chapter, we discuss non-symmetric random matrices, in particular, Ginibreensemble. Define an N ×N non-symmetric matrix GN = Gij1≤i,j≤N whoseentries are given by independent complex Gaussians:

Gij :=gRij +

√−1gIij√2

,

where gRij , gIij have density function 1√

2πe−

x2

2 , or equivalently, Gij is indepen-

dently distributed in C with density 1πe−|z|2 . GN is called (complex) Ginibre

Ensemble (GE). By the definition, GN is distributed on the N ×N complexmatrix space with density

p(G) =1

πN2 exp

(−

N∑i,j=1

|Gij|2)

=1

πN2 exp(−tr(G∗G)).

Note that in the equation of the density, the trace appears as in (1.1) and(1.2), and this form is suitable for applying the Schur decomposition. Letλ1, . . . , λN ∈ C be the eigenvalues of GN , and we refer to the Dyson’s way in[48, Section A.35]: We have a decomposition G = UTU∗, where U is an N×Nunitary matrix and T is an N ×N tridiagonal matrix whose diagonal entriesare the eigenvalues of GN . We rewrite T = D+S, where D = diag(λ1, . . . , λN)and the entries of S are i.i.d. standard Gaussians. Note that

tr(DS) = tr(SD) = 0,

which gives

tr(G∗G) = tr(UTU∗UT ∗U∗) = tr(DD∗) + tr(SS∗).

Combining the Jacobian∏

1≤i<j≤N |zi−zj|2 and integrating out the S variables,the jpdf of the eigenvalues has the following expression.

Theorem 1.2.1 (Ginibre, [28], [48]). The jpdf of the (unordered) complexeigenvalues of GN with respect to Lebesgue measure on CN is

p(z1, . . . , zN) =1

CN

exp

(−

N∑i=1

|zi|2) ∏

1≤i<j≤N

|zi − zj|2, (1.10)

where CN = πN∏N

j=1 j!.


We define the ESD of GN as

µ 1√NGN

:=1

N

N∑i=1

δ 1√Nλi.

Note that µ 1√NGN

is now a random probability measure on C. Mehta showed

that µ 1√NGN

converges to a uniform distribution on a unit disc in C by the

explicit form of (1.10), which convergence is called the circular law. To showthis, we rewrite the jpdf (1.10) by complex orthogonal polynomials, as in (1.8).We define the complex polynomials ϕn as

ϕn(z) :=zne−

|z|22

√n!π

, n ∈ 0 ∪ N.

These polynomials have the orthogonal property:∫ϕk(z)ϕℓ(z)dz = δkℓ, (1.11)

where dz = dRe(z)dIm(z). Using the property of the Vandermonde determi-nant in Remark 1.1.2 and the multilinearity of determinants, we find

exp

(−

N∑i=1

|zi|2) ∏

1≤i<j≤N

|zi − zj|2 = exp

(−

N∑i=1

|zi|2)

N

deti,j=1

zj−1i

N

deti,j=1

zij−1

= πN

N−1∏j=1

j!N

deti,j=1

ϕi−1(zj)N

deti,j=1

ϕi−1(zj) = πN

N−1∏j=1

j!N

deti,j=1

K(N)(zi, zj),

where

K(N)(zi, zj) :=N−1∑k=0

ϕk(zi)ϕk(zj). (1.12)

Combining the normalized constant CN , we conclude that the jpdf (1.10) hasthe expression:

p(z1, . . . , zN) =1

N !

N

deti,j=1

K(N)(zi, zj). (1.13)

We also define the k-correlation functions as

RN,k(z1, . . . , zk) :=N !

(N − k)!

∫p(z1, . . . , zN)

N∏i=k+1

dzi, 1 ≤ k ≤ N.

Using the orthogonal property (1.11) and Andreief’s identity, we have thefollowing.

1.2. GINIBRE ENSEMBLE 15

Proposition 1.2.2. [48, (15.1.29)] For any 1 ≤ k ≤ N ,

RN,k(z1, . . . , zk) =k

deti,j=1

K(N)(zi, zj).

As N → ∞, we obtain Rk(z1, . . . , zk) = detki,j=1KGin(zi, zj) with GinibreKernel

KGin(z, w) =1

πe−

|z|22

− |w|22

+zw, (1.14)

and we can construct a DPP using the above k-correlation function, see [35].Using Proposition 1.2.2, we deal with the ESD µ 1√

NGN

. We define the mean

density (or expectation of ESD) Eµ 1√NGN

as

< Eµ 1√NGN, f >:= E

[< µ 1√

NGN, f >

]= E[f(λ1)].

Note that the 1-correlation function gives the density function of λ1:

p(z) =1

NRN,1(z) =

1

πNe−|z|2

N−1∑k=0

|z|2k

k!, (1.15)

which implies as N → ∞,

p(z) =

1

π, |z| ≤ 1

0, |z| > 1

For detail estimates, see [48]. Therefore, the density of Eµ 1√NGN

converges to

µcir, where the limiting measure is the uniform distribution on the unit disc inC. Combining this convergence and moment calculations in [68, page 10], wehave

µ 1√NGN

→ µcir, a.s. (1.16)

Note that in more general cases, the above circular law also holds. Moreprecisely, Tao and Vu showed the following result under optimal moment as-sumptions.

Theorem 1.2.3. (Circular law, [60, Theorem 1.13]) Let XN = (Xij)1≤i,j≤N

be the N × N random matrix whose entries are i.i.d. random variables suchthat

E[Xij] = 0, E[|Xij|2] = 1.

Then,

µ 1√NXN

→ µcir, N → ∞, a.s.


-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

re_x

im_x

Figure 1.2. Plots of eigenvalues of GN scaled by N− 12 , N = 3000.

They also proved the universality principle of ESDs.

Theorem 1.2.4. (Universality principle, [60, Theorem 1.7]) Let X and Y becomplex random variables such that

E[X] = E[Y ] = 0, E[|X|2] = E[|Y |2] = 1.

Let XN = (Xij)1≤i,j≤N and YN = (Yij)1≤i,j≤N be N × N random matriceswhose entries Xij and Yij are i.i.d. copies of X and Y , respectively. Assumethat MN = (Mij)1≤i,j≤N is a deterministic matrix such that

supN

1

N2

N∑i,j=1

|Mij|2 <∞.

Let AN := XN +MN and BN := YN +MN . Then,

µ 1√NAN

− µ 1√NBN

→ 0, in prob.

Furthermore, if the ESD

µ( 1√NMN−zI)( 1√

NMN−zI)∗

converges to a limit for almost every z, then µ 1√NAN

− µ 1√NBN

→ 0, a.s.

1.3. ELLIPTIC GINIBRE ENSEMBLE 17

The proof of Theorem 1.2.4 is based on Girko’s identity, concentration in-equalities and the least singular value estimates. The terminology “universal”means that the convergence of ESDs does not depend on the distributions ofmatrix entries.

1.3 Elliptic Ginibre Ensemble

We define Elliptic Ginibre ensemble (EGE) as the linear combination oftwo N ×N GUEs:

J :=

√1 + τ√

2H1 +

√−1

√1 − τ√

2H2, −1 ≤ τ ≤ 1,

where H1 and H2 are independent and distributed in HN with density (1.2).EGE is one of the non-Hermitian random matrix models, and recently, manyapplications of non-Hermitian random matrices are discussed in physics: forexample, resonance scattering of quantum waves in open chaotic systems,quantum chromodynamics at non-zero chemical potential [1] and neural net-work dynamics [16]. We note that the matrix model used in the third exampleis very similar to EGE. This model was introduced as an interpolation betweenHermitian and non-Hermitian matrices in [58]. The parameter τ implies thedegree of Hermiticity. With τ = 1, the matrix J is Hermitian and thus GUE,and with τ = 0, it is completely non-Hermitian and thus GE, see Sections 1.1and 1.2. For −1 < τ < 1, J is distributed in the complex matrix space CN2

with density

p(J) ∝ exp

[− 1

1 − τ 2tr(JJ∗ − τ

2(J2 + J∗2))

].

Using the property of the trace and a similar deformation as the case of GE,the jpdf of the eigenvalues of J is described explicitly as

p(z1, . . . , zN) ∝ exp[− 1

1 − τ 2(

N∑i=1

|zi|2 −τ

2

N∑i=1

(z2i + z∗i2))]∏

i<j

|zi − zj|2

(1.17)

in [1, 41]. We define the ESD µ 1√NJN

as the same way in Section 1.2. Then,

the Girko’s elliptic law holds.

Theorem 1.3.1. (elliptic law, [30,58]) For fixed −1 < τ < 1,

µ 1√NJN

→ µelli, N → ∞, a.s.,

where µelli is the uniform distribution on the ellipsez ∈ C ;

(Re(z)1+τ

)2+(

Im(z)1−τ

)2≤ 1

.


The condition −1 < τ < 1 is called “strong non-Hermiticity” since theanti-Hermitian part

√−1

√1−τ√2H2 of J is the same order in N as the Hermitian

part. In contrast, the limiting behavior under the condition of “weak non-Hermiticity” is also known. Suppose that 1 − τ = α2

Nby some α > 0. Then

in the limit N → ∞, the mean density, that is, density of Eµ 1√NJN

behaves

asymptotically as

p(z) ∼ psc(x)p(y), z = x+√−1y,

where

psc(x) =1

2π

√4 − x21[−2,2], Wigner′s semicircle distribution,

p(y) =

√N2

2πα2exp(−N

2y2

2α2), Gaussian distribution.

This result is first observed in [25] by perturbation theory, but more detailedestimates are in [1, 3]. Another study was given by Akemann, Cikovic andVenker in [2], using correlation functions and kernels. They introduced thefollowing ensemble distributed on CN2

with density

pN,tr2(J) :=1

ZN,tr2exp

[− N

1 − τ 2tr(JJ∗ − τ

2(J2 + J∗2)) − γ(tr(JJ∗) −NKp)

2

],

where γ ≥ 0, Kp ∈ R are fixed, −1 < τ < 1. In the above density, the termγ(tr(JJ∗) − NKp)

2 implies the fixed trace ensemble model, and with γ = 0,we recover the normalized EGE defined in this section. Let Rtr2

N,k(z1, . . . , zk)be the k-correlation functions of eigenvalues obtained by the ensemble. Then,we have the following claim.

Theorem 1.3.2. (Limit of strong non-Hermiticity, [2, Theorem 1 (b)]) Let−1 < τ < 1 be fixed. Then there exist constants C, c1, c2 > 0, depending only

on Kp, τ and γ, such that with E :=z ∈ C ; c1Re(z)2 + c2Im(z)2 ≤ 1

the

following holds. For k ∈ N, z ∈ E, z1, . . . , zk ∈ C, as N → ∞,

1

(CN)kRtr2

N,k

(z +

z1√CN

, . . . , z +zk√CN

)=

k

deti,j=1

KGin(zi, zj) +O

(1√N

),

where KGin(z, w) is given by (1.14).

The above theorem tells us that Ginibre kernel KGin(z, w) is universal; thelimiting kernel does not depend on the parameter Kp, τ and γ, whereas thelimiting ESD of eigenvalues converges to the uniform distribution on the ellipseE. For the weak non-Hermiticity limit, we also have the following.

1.4. EIGENVECTOR CORRELATIONS 19

Theorem 1.3.3. (Limit of weak non-Hermiticity, [2, Theorem 3 (b)]) Thereexists a constant C > 0, depending only on Kp and γ, such that the following

holds. Define ν(x) := C2π

√4C− x2 and τ = τN := 1 − α2

2Nν(x)2, α > 0. Then for

k ∈ N, z1, . . . , zk ∈ C, as N → ∞,

1

(Nν(x))2kRtr2

N,k

(x+

z1Nν(x)

, . . . , x+zk

Nν(x)

)=

k

deti,j=1

Kweak(zi, zj) +O

(logN√N

),

where for zi := xi +√−1yi,

Kweak(z1, z2) :=

√2

πα2exp

[−y

21 + y22α2

]1

2π

∫ π

−π

exp

[−α

2u2

2+√−1u(z1 − z2)

]du.

In the limit α → 0, we have the Dyson-Gaudin-Mehta correlation kernel,that is, the sine kernel. For any continuous and compactly supported functionf : Ck → R,

limα→0

∫Ck

f(z1, . . . , zk)k

deti,j=1

Kweak(zi, zj)dz1 . . . dzk

=

∫Rk

f(x1, . . . , xk)k

deti,j=1

sin(π(xi − xj))

π(xi − xj)dx1 . . . dxk.

1.4 Eigenvector correlations

For non-normal matrices, the overlaps also have been studied. They are alsocalled eigenvector correlations or condition numbers. Let M be an N×N non-normal matrix that has distinct eigenvalues λ1, . . . , λN . In general, eigenvaluesare complex variables. Then, there exist the right eigenvectors Rj and lefteigenvectors Lj, j = 1, . . . , N such that

MRj = λjRj,tLjM = λj

tLj.

Since Rj and Lj have a biorthogonal relation, we normalize them as

tLiRj = δij.

In these settings, the overlaps of M are defined by

Oij := (R∗jRi)(L

∗jLi).

Clearly, if M is normal, then by the orthogonality of eigenvectors, Oij = δij,and so there is no interest; on the contrary, for non-normal cases, they playan important role. The non-orthogonality of eigenvectors affects the behaviorof eigenvalues, see [62]. In the case of GE, an early observation was given


by Chalker and Mehlig in [14, 15]. They estimated the asymptotic behaviorof the conditional expectations of overlaps. For normalized GE GN√

Nand any

|z1|, |z2| < 1,

E[O11|λ1 = z1] ∼ N(1 − |z1|2), (1.18)

E[O12|λ1 = z1, λ2 = z2] ∼ − 1

N

1 − z1z2|z1 − z2|4

1 − (1 +N |z1 − z2|2)e−N |z1−z2|2

1 − e−N |z1−z2|2,

(1.19)

as N → ∞. They found that the conditional expectations (1.18) and (1.19)have expressions of determinants of tridiagonal matrices and that of five-diagonal matrices, respectively. These facts are useful to calculate the condi-tional expectations because these determinants have recursion formulas. Theestimation (1.18) tells us the typical order of the diagonal overlap O11 = O(N),and recently, Bourgade and Dubach showed the limiting conditional distribu-tion.

Theorem 1.4.1. (Limiting distribution of diagonal overlaps, [10, Theorem1.1]) Let κ > 0 be an arbitrary small constant. Then, uniformly in |z| <1 −N− 1

2+κ, for any continuous bounded function f ,

E

[f

(O11

N(1 − |z|2)

)|λ1 = z

]→∫ ∞

0

f(t)e−

1t

t3dt, N → ∞.

Here, the limiting distribution in the right hand side is the same distribu-tion of 1

γ2, where γα has density 1

Γ(α)xα−1e−x

1[0,∞). Moreover, they also proved

the estimate in (1.19) rigorously as following.

Theorem 1.4.2. (Expectation of off-diagonal overlaps, [10, Theorem 1.3])For any κ ∈ (0, 1

2) and small ϵ > 0, uniformly in z1, z2 such that |z1| <

1 −N− 12+κ, ω :=

√N |z1 − z2| < Nκ−ϵ, we have

E[O12|λ1 = z1, λ2 = z2] = −N 1 − z1z2|ω|4

1 − (1 + |ω|2)e−|ω|2

1 − e−|ω|2 (1 +O(N−2κ+ϵ)).

Theorem 1.4.2 says that in the mesoscopic regime |ω| → ∞, we have

E[O12|λ1 = z1, λ2 = z2] = − 1 − z1z2N |z1 − z2|4

(1 + o(1)),

which implies that the expectation of off-diagonal overlaps decreases with theseparation of eigenvalues. These results give the diffusive exponent of theeigenvalues processes of GE, discussed in Section 2.3. We remark that Fyo-dorov also showed a similar result for real Ginibre ensemble with supersymmet-ric approach in [24]. Furthermore, the result for real EGE was also reportedin [26].

Chapter 2

Matrix-valued processes

In this Chapter, we consider matrix-valued processes and eigenvalue processesbased on matrix models introduced in Chapter 1. Dyson’s Brownian motionmodels are typical models of time evolution for random matrices. In Sections2.1 and 2.2, we mainly refer to normal matrix-valued processes, in particular,Hermitian, unitary and positive definite matrices. We remark that the rela-tion between eigenvalue processes of these models and non-colliding Brownianmotions are also discussed. In Section 2.3, we refer to non-symmetric (non-normal) matrix-valued processes, in particular, the time evolution model ofGinibre ensemble. In the discussion, the related interacting particle systemsare also considered, but contrary to normal matrix cases, these models arevery different from eigenvalue processes of Ginibre ensemble.

2.1 Dyson’s Brownian motions

Dyson developed Gaussian ensembles to a dynamical system according to theCoulomb gas model in [19]. He considered eigenvalue processes of Hermi-tian matrices as Brownian motions with “repulsive” force on each other. Weexplain the above eigenvalue processes, which are called Dyson’s Brownianmotions. For β = 1, 2, 4, define the N×N Hermitian matrix-valued processesH(β)(t) = (H

(β)kℓ (t))1≤k,ℓ≤N as following. For β = 1, 2,

H(β)kℓ (t) :=

√

2

βBkk(t), k = ℓ,

BRkℓ(t) +

√−1(β − 1)BI

kℓ(t)√β

, k < ℓ.

21

22 CHAPTER 2. MATRIX-VALUED PROCESSES

For β = 4,

H(4)kℓ (t) :=

1√2Bkk(t), k = ℓ,

B(1)kℓ (t) + iB

(2)kℓ (t) + jB

(3)kℓ (t) + kB

(4)kℓ (t)

2, k < ℓ.

Here, Bkk, BRkℓ, B

Ikℓ, B

(a)kℓ , a = 1, . . . , 4 are independent one-dimensional Brow-

nian motions and i, j, k are quaternions. Clearly, the parameter β impliesthe matrix symmetry and corresponds to GOE, GUE and GSE as β = 1, 2, 4,respectively. Let λλλ(t) = (λ1(t), . . . , λN(t)) be the real eigenvalue process ofH(β)(t). Then, Dyson proved the following theorem.

Theorem 2.1.1. (Dyson’s Brownian motions, [19]) For β = 1, 2, 4, the eigen-value process λλλ(t) ofH(β)(t) satisfies the stochastic differential equations (SDEsfor short) :


2

∑j(=i)

1

λi(t) − λj(t)dt, i = 1, . . . , N, (2.1)

where Bi, i = 1, . . . , N , are independent one-dimensional Brownian motions.

Note that in the definition of H(β)(t), if we replace Brownian motions ofthe entries by Ornstein-Uhlenbeck processes

dX(t) = dB(t) − X(t)

2dt,

then the SDEs (2.1) have the Ornstein-Uhlenbeck drifts:


2

∑j(=i)

1

λi(t) − λj(t)dt− β

2

λi(t)

2dt,

so that the eigenvalue processes satisfying the above SDEs have stationarydistribution (1.10). Dyson derived the SDEs of the eigenvalues (2.1) by per-turbation theory, but of course, the above SDEs are also obtained by stochasticanalysis, for example, see [4, Section 4.3]. Rogers and Shi proved that stochas-tic processes, which satisfy more general SDEs, are non-colliding:

Tcol := inft > 0; λi(t) = λj(t) for some i = j = ∞, a.s. (2.2)

This fact is naturally expected because the eigenvalues exert “repulsive” forceon each other as seen in Section 1.1, and also see [59]. We remark that theSDEs (2.1) also appear in a system of non-colliding diffusions. Katori andTanemura constructed a system of N non-colliding Brownian motions and

2.2. SOME NORMAL MATRIX-VALUED PROCESSES 23

proved that this system satisfies the same SDEs (2.1) in [38, 39]. Their claimjustifies Dyson’s arguments of repulsive forces. Another study of Dyson’smodels, infinite dimensional SDEs related to (2.1) are discussed. For β ≥ 1, astrong solution of the following infinite dimensional SDE

dXi(t) = dBi(t) +β

2limk→∞

∑j:|j−i|≤k

1

Xi(t) −Xj(t)dt, i ∈ Z (2.3)

exists, and under suitable initial configurations, the pathwise uniqueness ofthe solution holds in [63, Theorem 1.2]. Moreover, a convergence of the finiteSDE (2.1) to the infinite SDE (2.3) was proved in the same paper [63].

2.2 Some normal matrix-valued processes

2.2.1 Unitary matrix-valued processes

Circular orthogonal, unitary and symplectic ensembles (COE, CUEand CSE, respectively) are introduced in [18]. These ensembles are N × Nrandom matrices and distributed on the orthogonal group O(N), the unitarygroup U(N) and the symplectic group Sp(N) with each of the Haar measure.The eigenvalues of the three ensembles are on the unit circle, and so we denotethem by eiθj , j = 1, . . . , N . Then, the jpdf of random angles θj ∈ [0, 2π], j =1, . . . , N with respect to the Lebesgue measure on [0, 2π]N is given by

p(θ1, . . . , θN) =1

C(β)N

∏1≤j<k≤N

|eiθj − eiθk |β,

where C(β)N is a normalized constant and β = 1, 2, 4 correspond to COE, CUE

and CSE, respectively, see in [48, Chapter 9]. Dyson applied a similar analogymentioned in Section 2.1 and constructed the interacting Brownian motionson the unit circle. More precisely, he derived the SDEs of eigenvalue processesof the time evolution of the above three ensembles as

dθj(t) =√

2dBj(t) +β

2

∑k(=j)

cot

(θj(t) − θk(t)

2

)dt, j = 1, . . . , N,

where β = 1, 2, 4 correspond to COE, CUE and CSE, respectively. Related tothis model, non-colliding Brownian motions on the unit circle were reportedin [42].

2.2.2 Wishart processes

Bru derived eigenvalue processes of positive definite Hermitian matrix which iscalled Wishart processes [11, 12]. We define an N ×M matrix-valued process


B(t) whose entries are independent one-dimensional Brownian motions and

W (t) := tB(t)B(t).

W (t) is an M × M positive definite Hermitian matrix-valued process, andλ1(t) ≥ · · · ≥ λM(t) ≥ 0 denote the eigenvalues of W (t). Here, we assumeN ≥M and λ1(0) > · · · > λM(0). Then we have the following.

Theorem 2.2.1. (Bru, [11, 12]) The eigenvalue processes of W (t) satisfy theSDEs:

dλi(t) = 2√λi(t)dBi(t) +Ndt+

∑k(=i)

λi(t) + λk(t)

λi(t) − λk(t)dt, i = 1, . . . ,M, (2.4)

where Bi, i = 1, . . . ,M , are independent one-dimensional Brownian motions.Moreover, the eigenvalues do not collide with each other, that is, Tcol = ∞, a.s.

She also derived the SDEs of eigenvectors.

Theorem 2.2.2. [11, Theorem 2]. W (t) can be diagonalized by an orthogonalmatrix-valued process U(t):

U(t)∗W (t)U(t) = diag(λ1(t), . . . , λM(t)).

Here, U(t) is a continuous martingale whose entries satisfy the SDEs:

duij(t) =∑k(=j)

uik(t)

√λk(t) + λj(t)

(λk(t) − λj(t))2dβkj(t) −

uij(t)

2

∑k( =j)

λk(t) + λj(t)

(λk(t) − λj(t))2dt,

where βij, 1 ≤ i < j ≤M , are independent one-dimensional Brownian motionsand independent of λi, i = 1, . . . ,M .

We remark that the non-colliding argument was proved by using the func-tion f : (x1, . . . , xM) ∈ RM ;x1 > · · · > xM → R defined as

f(x1, . . . , xM) :=∑i<j

log(xi − xj),

and law of the iterated logarithm for Brownian motions. This method wasfound in [43, 52], but the choice of such a function depends on the SDEseigenvalues satisfy.

For M = 1, the process W (t) =∑N

i=1 |bi(t)|2, where bi, i = 1, . . . , N areindependent one-dimensional Brownian motions, is a squared Bessel process;W (t) satisfies the following SDE:

dX(t) = 2√X(t)db(t) +Ndt, (2.5)

2.2. SOME NORMAL MATRIX-VALUED PROCESSES 25

where b is a one-dimensional Brownian motion [56, Chapter 11]. In the pointof view, for M ≥ 1, the Wishart process W (t) is the generalization of squaredBessel processes. We also note that the eigenvalue processes of W (t) satisfying(2.4) are studied as non-colliding diffusion systems in [33,40].

At the end of Subsection, we refer to the generalized result for derivingSDEs of eigenvalues of Hermitian matrix-valued processes in [40]. This re-sult means that for given Hermitian matrix-valued processes whose entries arecontinuous semi-martingale, we obtain the SDEs which the eigenvalues satisfy.Assume that an N×N Hermitian matrix-valued process H(t) = (hij(t))1≤i,j≤N

has continuous semi-martingale entries. U(t) denotes the family of N×N uni-tary matrices which diagonalizes H(t):

U(t)∗H(t)U(t) = diag(λ1(t), . . . , λN(t)) =: Λ(t),

where λ1(t), . . . , λN(t) are eigenvalues of H(t) in the increasing order. Weoften use Ito’s calculus for semi-martingale matrices to obtain the SDEs ofeigenvalues as such a form:

dH = d(UΛU∗) =dUΛU∗ + UdΛU∗ + UΛdU∗

+ dUdΛU∗ + dUΛdU∗ + UdΛdU∗. (2.6)

Define Γij(t), 1 ≤ i, j ≤ N as

Γij(t)dt = (U(t)∗dH(t)U(t))ij(U(t)∗dH(t)U(t))ji

By using the calculus in (2.6), the following result holds.

Theorem 2.2.3. (Generalized Bru’s Theorem, [40]) The eigenvalue processλλλ(t) = (λ1(t), . . . , λN(t)) satisfies the SDEs:

dλi(t) = dMi(t) + dJi(t)dt, i = 1, . . . , N

where the martingale Mi(t) has the quadratic variation d⟨Mi⟩t = Γii(t)dt andJi(t) is the process with finite variation given by

dJi(t) =N∑j=1

1

λi(t) − λj(t)1λi(t)=λj(t)Γij(t)dt+ dΥi(t),

where dΥi(t) is the finite variation part of (U(t)∗dH(t)U(t))ii.

The above theorem gives the SDEs (2.1) and (2.4) by independence ofmatrix entries, and several modifications of eigenvalue processes are discussedin [40].


2.3 Time-depending Ginibre Ensemble and Over-

laps

The study of matrix-valued process for non-normal matrices is lesser than thatfor normal case since the eigenvectors and overlaps should be also concernedtogether as mentioned in Section 1.4. Nevertheless, there are remarkable re-sults for GE. This model is non-symmetric matrix-valued process whose entriesare given by independent complex Brownian motions. Firstly, we refer to theresult by Grela and Warcho l in [13, 34]. Consider an N × N non-symmetricmatrix-valued process G(t) = (Gij(t))1≤i,j≤N whose entries are given by

Gij(t) :=BR

ij(t) +√−1BI

ij(t)√2

,

where BRij , B

Iij are independent one-dimensional Brownian motions, so that the

complex quadratic variations are

d⟨Gij, Gkℓ⟩t = 0, d⟨Gij, Gkℓ⟩t = δikδjℓdt. (2.7)

Here, for complex semi-martingales M(t) = MR(t) +√−1M I(t) and N(t) =

NR(t) +√−1N I(t), the complex quadratic variation is defined as

d⟨M,N⟩t := d⟨MR, NR⟩t − d⟨M I , N I⟩t +√−1(d⟨MR, N I⟩t + d⟨M I , NR⟩t

).

(2.8)

To obtain eigenvalue processes of G(t), we apply a similarity transformation:

G(t) = X(t)Λ(t)X(t)−1,

where Λ(t) = diag(λ1(t), . . . , λN(t)) and X(t) = (R1(t), . . . , RN(t)) is the righteigenvector matrix such that

G(t)Rj(t) = λj(t)Rj(t),

as in Section 1.4. Note that if we take the left eigenvectors Lj(t), j = 1, . . . , Nof G(t) such that

X(t)−1 =

tL1(t)

...tLN(t)

,

then tLi(t)Rj(t) = δij. By Ito’s calculus for semi-martingale matrices in (2.6),we formally find

δG(t) = δX(t)Λ(t) + dΛ(t) + Λ(t)δX ′(t)

2.3. TIME-DEPENDING GINIBRE ENSEMBLE AND OVERLAPS 27

+ δX(t)dΛ(t) + dΛ(t)δX ′(t) + δX(t)Λ(t)δX ′(t),

where

δG(t) = X(t)−1dG(t)X(t), δX(t) = X(t)−1dX(t), δX ′(t) = dX(t)−1X(t).

By the calculations in [34, Appendix A], we obtain the infinitesimal changesof eigenvalues and eigenvectors as following:

dλi(t) =(δG(t))ii +∑k(=i)

(δG(t))ik(δG(t))kiλi(t) − λk(t)

,

(δX(t))ij =(δG(t))ij

λj(t) − λi(t)+∑k(=j)

(δG(t))ik(δG(t))kj(λi(t) − λj(t))(λk(t) − λj(t))

− (δG(t))ij(δG(t))jj(λi(t) − λj(t))2

, i = j,

(δX(t))ii =0.

(2.9)

We use a notation of the complex quadratic variation in (2.8) as dM(t)dN(t) :=d⟨M,N⟩t. In the notation, by the equation (2.7) we have

(δG(t))ij(δG(t))kℓ = 0,

(δG(t))ij(δG(t))kℓ = (X(t)−1dG(t)X(t))ij(X(t)−1dG(t)X(t))kℓ

= (X(t)∗X(t))ℓj(X(t)∗X(t))−1ik dt

=: (A(t))ℓj(A(t)−1)ikdt,

where A(t) := X(t)∗X(t). This gives time-depending overlaps of G(t):

Oij(t) = (A(t)−1)ij(A(t))ji.

Using the above quadratic variations, we find that in (2.9), the products of(δG(t))ij vanish, and we get

dλi(t) =(δG(t))ii,

(δX(t))ij =(δG(t))ij

λj(t) − λi(t), i = j,

(δX(t))ii =0.

(2.10)

Note that in the formal calculations, the SDEs of eigenvalues of G(t) haveno drift terms; the eigenvalue processes of G(t) are completely different fromDyson’s Brownian motions as in (2.1), and the complex quadratic variationsof the processes are described by their overlaps:

d⟨λi, λj⟩t = (δG(t))ii(δG(t))jj = (A(t))ji(A(t)−1)ijdt = Oij(t)dt. (2.11)


The above fact is also showed in [10, Appendix A] rigorously by a simi-lar calculation. Furthermore, by Cauchy-Schwarz inequality we can showOii(t) ≥ 1, which tells us that the motion of each of eigenvalues is faster thana two-dimensional Brownian motion because they have the complex quadraticvariation d⟨λi, λi⟩t = Oii(t)dt. Together with (2.11), the non-zero complexquadratic variations in (2.10) are summarized as following:

dλi(t)dλj(t) = (A(t))ji(A(t)−1)ijdt,

(dX(t))kℓdλi(t) = (A(t))iℓ∑n(=ℓ)

(X(t))kn(A(t)−1)niλℓ(t) − λn(t)

dt,

(dX(t))kℓ(dX(t))nm =∑α(=ℓ)β( =m)

(X(t))kα(X(t))nβ(A(t))mℓ(A(t)−1)αβ

(λℓ(t) − λα(t))(λm(t) − λβ(t)).

(2.12)

In [34, Chapter III], a solution of Fokker-Planck equation of the jpdf of eigenval-ues and eigenvectors of G(t) was derived. Using the notation dR(t) ≡ δX(t) =X(t)−1dX(t), we rewrite (2.12) as

dλi(t)d(R(t))kℓ =(A(t))ℓi(A(t)−1)ik

λℓ(t) − λk(t)dt,

d(R(t))ijd(R(t))kℓ =(A(t))ℓj(A(t)−1)ik

(λj(t) − λi(t))(λℓ(t) − λk(t))dt.

(2.13)

For a C2 function f , which belongs to a suitable domain, we obtain the fol-lowing SDE of f(R(t),Λ(t)) by Ito’s formula:

dft = dfvft + dmft,

where the differential operators dfv and dm are

dfv :=∑i,j

dλidλj∂λi,λj+∑i,k =ℓ

dλidRkℓ∂λi,Rkℓ+∑i,k =ℓ

dλidRkℓ∂λi,Rkℓ

+∑k =ℓn=m

dRkℓdRnm∂Rkℓ,Rnm,

dm :=∑i

dλi∂λi+∑i

dλi∂λi+∑k =ℓ

dRkℓ∂Rkℓ+∑k =ℓ

dRkℓ∂Rkℓ.

Here, we omit the time parameter t. By a usual observation and rewriting Ras X, we formally get the Fokker-Planck equation of Pt:

∂tPt =∑i,j

∂λi,λj

(dλidλjdt

Pt

)+∑i,k,ℓ

∂λi,Xkℓ

(dλidXkℓ

dtPt

)


+∑i,k,ℓ

∂λi,Xkℓ

(dλidXkℓ

dtPt

)+∑

k,ℓ,n,m

∂Xkℓ,Xnm

(dXkℓdXnm

dtPt

), (2.14)

where Pt = Pt(X,Λ). By using the equations (2.12), we solve the aboveequation (2.13) under the following separated form:

Pt(X,Λ) = F (Λ)Qt(X,Λ). (2.15)

If we take F ≡∑

i<j |λi−λj|4, then we can simplify Qt satisfying the equations(2.14) and (2.15). The above choice of F is reasonable in [34, Appendix B].Therefore, we find a solution of the Fokker-Planck equation of Pt in (2.14) as

Pt(X,Λ)dΛdR =1

(2πt)N2N !

∑i<j

|λi − λj|4

× exp

−1

t

∑i,j,k,ℓ

(X∗−1ik λkX

∗kj −X

(0)∗−1ik λk

(0)X

(0)∗kj )

× (XjℓλℓX−1ℓi −X

(0)jℓ λ

(0)ℓ X

(0)−1ℓi )

dΛdR,

(2.16)

where dΛ =∏

i dλidλi, dR =∏

i =j dRij, dR = X−1dX, and λ(0), X(0) are

the initial condition. If we take X(0) = 0, then we have the solution of (2.14)formally as

Pt(X,Λ; X(0) = 0)dΛdR =1

(2πt)N2N !

∑i<j

|λi − λj|4 exp

−1

t

∑i,j

Oijλiλj

dΛdR,

(2.17)

found in [34, equation (40)].Secondly, we refer to the results by Bourgade and Dubach in [10]. Replaceentries of G(t) as normalized independent Ornstein-Uhlenbeck processes:

dGij(t) =BR

ij(t) +√−1BI

ij(t)√2N

− 1

2Gij(t)dt, (2.18)

so that the stationary measure of eigenvalues of G(t) is (1.10) in Section 1.2.Then, by a similarity transformation and Ito’s formula for semi-martingalematrices as we see in the above discussion, we have the following.

Proposition 2.3.1. [10, Proposition A.1]The eigenvalue process λλλ(t) = (λ1(t), . . . , λN(t)) of G(t) defined by (2.18) is asemi-martingale and satisfies the SDEs:

dλi(t) = dMi(t) −1

2λi(t)dt, (2.19)


where the martingales Mi, i = 1, . . . , N are characterized by

d⟨Mi,Mj⟩t =Oij(t)

Ndt.

Note that the above claim is shown in (2.10) and (2.11). In Chapter 3, wegive explicit SDEs of the above equation (2.19) for EGE, which includes theabove time-depending GE model. Moreover, combining the results of staticGE, Theorem 1.4.1 and Theorem 1.4.2 in Section 1.4, Bourgade and Dubachalso show the diffusive exponent of eigenvalue processes of G(t) defined by(2.18).

Theorem 2.3.2. (Diffusive exponent, [10, Corollary 1.6]) Let c, a > 0 bearbitrary small constant. Assume that entries of G(0) are i.i.d. random vari-ables with density N

πe−N |z|2 with respect to the Lebesgue measure on C. Let

B ⊂ z ∈ C; |z| < 1 − c be a ball, t < N−c. Then as N → ∞,

E[|λ1(t) − λ1(0)|2, λ1(0) ∈ B

]= t

∫B(1 − |z|2)dRezdImz

π(1 + o(1)),

E[(λ1(t) − λ1(0))(λ2(t) − λ2(0)), λ1(0) ∈ B ∩ |λ1(0) − λ2(0)| < N−a

]= o(tN−2a).

From the time scale result in the second equation of Theorem 2.3.2, theyexpect that conditionally on λi(0) = z, for N−1+c < t < N−c, the process

(λi(ts) − z)0≤s≤1√t(1 − |z|2)

converges in distribution to a complex Brownian motion. Moreover, from theresult in the third equation of Theorem 2.3.2, they also expect that theselimiting processes associated to different eigenvalues are independent.

Finally, we remark that the time evolution of GE, the eigenvalue processesof G(t), is very different from the Dyson’s Brownian motion models, that is, thelog-gas analogy.　 A construction of N two-dimensional interacting particlesystems associated to a planar coulomb gas is given in [8]. In particular, bytaking a potential function as

H(z) =1

N

N∑i=1

|zi|2 −1

N2

∑1≤i<j≤N

log |zi − zj|2,

Bolley, Chafaı and Fontbona studied a diffusion process XN = (X i.N)1≤i≤N

which is a solution of the following SDEs:

dX i,Nt =

√2αN

βNdBi,N

t − 2αN

NX i,N

t dt− 2αN

N2

∑j(=i)

X i,Nt −Xj,N

t

|X i,Nt −Xj,N

t |2dt, 1 ≤ i ≤ N,


where Bi,N , 1 ≤ i ≤ N are two-dimensional independent Brownian motionsand αN > 0, βN = N2. They proved that pathwise uniqueness and strongexistence hold for the above SDEs in [8, Theorem 1.1]. Another construction ofparticle systems associated to GE is studied by Osada in [53]. He constructedan infinite interacting particle system X = (X i)i∈N in a two-dimensional space,which is described by the following infinite SDE:

dX it = dBi

t + limr→∞

∑|Xi

t−Xjt |<r, j =i

X it −Xj

t

|X it −Xj

t |2dt, i ∈ N, (2.20)

where Bi, i ∈ N are two-dimensional independent Brownian motions. Theconstruction of such an infinite interacting particle system is given by usingGinibre kernel (1.14) and Dirichlet form approach. He proved the existence ofa solution of (2.20) and another representation of the above infinite SDEs:

dX it = dBi

t −X itdt+ lim

r→∞

∑|Xj

t |<r, j =i

X it −Xj

t

|X it −Xj

t |2dt, i ∈ N.

As seen above, constructions of finite or infinite particle systems related to GEare studied, based on the density of eigenvalues of GE in (1.10), with differ-ent approach. We emphasis that the eigenvalue processes of GE obtained bya non-symmetric matrix-valued process and stochastic processes constructedby interacting particle systems as mentioned above are essentially different,although in Dyson’s models, they agree on each other. This claim is verifiedin Chapter 3 by describing the explicit SDEs of eigenvalue processes of EGE.

Chapter 3

Matrix-valued processassociated with Elliptic GinibreEnsemble

In this Chapter, we consider the time evolution model of Elliptic Ginibreensemble (EGE) and show our main results and proofs in [74]. The main resultis to give the stochastic differential equations (3.4) with parameter −1 ≤ τ ≤ 1that the eigenvalue processes of EGE satisfy. Recall that Dyson’s Brownianmotion models are discussed in Section 2.1, and when β = 2, we call simplythis Dyson’s model in this Chapter. On the basis of observations in Chapters1 and 2, we consider the non-normal matrix-valued process of EGE whoseentries are given by independent Brownian motions. This model naturallygives an interpolation between Dyson’s model and the time evolution of GEby using Hermiticity parameter τ in the same way as the static case in Section1.3. In the main theorem (Theorem 3.1.1), we show that for −1 ≤ τ ≤ 1, theeigenvalue processes of EGE satisfy the stochastic differential equations whichhave the drift of Dyson’s model except for τ = 0, and we also show the explicitform of their time-depending overlaps described by given Brownian motions.As a result, we obtain the complex martingales of the eigenvalue processes ofGE explicitly, which tell us the interaction of the eigenvalues by the form ofdifference product, that is, Vandermonde determinant. In the case of 2 × 2matrix, we can show that the quadratic variation of the diagonal overlap andthe distance of the two eigenvalues is negative by using our explicit form,which proves the fact in [7]; as the two eigenvalues get closer to each other,they move faster. We also show for −1 ≤ τ ≤ 1, the eigenvalue processes ofEGE are non-colliding.

32

3.1. SETTINGS AND MAIN RESULTS 33

3.1 Settings and Main Results

We consider the N ×N matrix-valued process for EGE and define this modelas follows:

J(t) :=

√1 + τ√

2H1(t) +

√−1

√1 − τ√

2H2(t), t ≥ 0, −1 ≤ τ ≤ 1, (3.1)

where

(H1(t))kℓ :=

Bkk(t) k = ℓ

BRkℓ(t) +

√−1BI

kℓ(t)√2

k < ℓ

(H1(t))ℓk k > ℓ

, (H2(t))kℓ :=

bkk(t) k = ℓ

bRkℓ(t) +√−1bIkℓ(t)√2

k < ℓ

(H2(t))ℓk k > ℓ

.

Here, Bkk, BRkℓ, B

Ikℓ, bkk, b

Rkℓ, b

Ikℓ (1 ≤ k ≤ ℓ ≤ N) are independent one-dimensional

Brownian motions defined on a filtered probability space (Ω,F , Ftt≥0,P).The entries of J(t) have the correlation described by complex quadratic vari-ations:

d⟨Jij, Jkℓ⟩t = δikδjℓdt, d⟨Jij, Jkℓ⟩t = τδiℓδjkdt. (3.2)

Recall that the complex quadratic variation is defined in (2.8). The quantities(3.2) express the Hermiticity of the matrix J(t) by τ . By construction of J(t),we get Dyson’s model with τ = 1 and Ginibre dynamics with τ = 0. J(t)has pure imaginary eigenvalue processes with τ = −1, which are just Dyson’smodel on the imaginary axis. Thus it is essential to consider 0 ≤ τ ≤ 1.From the perspective of normality of matrices, the case of τ = 1 and τ =0 are extreme; in the former case each of the eigenvalue processes has thedrift term, similar to that in (2.1), which takes a larger absolute values asit gets closer to the other eigenvalues, and in the latter case the eigenvalueprocesses are complex martingales. This fact is pointed out in Section 2.3.We denote the eigenvalue processes of J(t) by λλλ(t) = (λ1(t), . . . , λN(t)). Asmentioned above, these processes usually take complex values, so we writeλi(t) = λRi (t) +

√−1λIi (t), i = 1, . . . , N . We assume the following initial

condition:

J(0) has simple spectrum, that is, λ1(0), . . . , λN(0) are distinct. (3.3)

We denote the N × N identity matrix by I, and for a square matrix A, wedefine the (N − 1) × (N − 1) minor matrix Ak|ℓ that is obtained by removingthe k-th row and the ℓ-th column from A. Then we have the following result.

Theorem 3.1.1. [74, Theorem 2.1] For −1 ≤ τ ≤ 1 with initial condition(3.3), the eigenvalue process λλλ(t) = (λ1(t), . . . , λN(t)) of J(t) is complex semi-

34 CHAPTER 3. MATRIX-VALUED PROC. ASSOC. WITH EGE

martingale and satisfies the stochastic differential equations:

dλi(t) =N∑

k,ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ)∏j(=i)(λi(t) − λj(t))

dJkℓ(t) + τ∑j(=i)

1

λi(t) − λj(t)dt,

t ≥ 0, i = 1, . . . , N,

(3.4)

and the real quadratic variations are given by

d⟨λRi ⟩t =Oii(t) + τ

2dt, d⟨λIi ⟩t =

Oii(t) − τ

2dt, d⟨λRi , λIi ⟩t = 0, (3.5)

d⟨λRi , λRj ⟩t = d⟨λIi , λIj⟩t =Re(Oij(t))

2dt,

−d⟨λRi , λIj⟩t = d⟨λIi , λRj ⟩t =Im(Oij(t))

2dt, i = j

(3.6)

with

Oij(t) :=

∑Nk=1 det

(((λi(t)I − J(t))(λj(t)I − J(t))∗

)k|k

)∏

p(=i)(λi(t) − λp(t))∏

q(=j) (λj(t) − λq(t)). (3.7)

Moreover, the eigenvalues do not collide with each other, that is, Tcol = ∞, a.s.,as defined in (2.2).

By (3.5) and (3.6), the complex quadratic variations of λλλ(t) are describedas d⟨λi, λj⟩t = Oij(t)dt. Although we do not use the non-orthogonality ofeigenvectors in our proof, the quadratic variations coincide with the overlapsof J(t) because the result and proof for the eigenvalue processes of GE in [10]are also valid for our model. For this reason, we use the notation Oij(t) forthe quadratic variations.

Remark 3.1.2. For matrix entries of J(t), if we replace Brownian motionswith Ornstein-Uhlenbeck processes

d(H1(t))kk = dBkk(t) − 1

2(H1(t))kkdt,

d(H1(t))kℓ =dBR

kℓ(t) +√−1dBI

kℓ(t)√2

− 1

2(H1(t))kℓdt, k < ℓ,

and also for entries of H2(t), we obtain the Ornstein-Uhlenbeck drift in theSDE (2.19). Indeed, by the explicit calculus in Subsections 3.2.1 and 3.2.2, wehave

dλi(t) =N∑

k,ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ)∏j( =i)(λi(t) − λj(t))

dMkℓ(t)

3.1. SETTINGS AND MAIN RESULTS 35

− 1

2

N∑k,ℓ=1


Jkℓ(t)dt+ τ∑j(=i)

1


where Mkℓ(t) are martingale terms of Jkℓ(t). For each integer k, by the cofactorexpansion of the k-th row we obtain

det(λi(t)I − J(t))

=N∑ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ)(λi(t)δkℓ − Jkℓ(t))

= λi(t) det((λi(t)I − J(t))k|k) −N∑ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ)Jkℓ(t) = 0.

Taking a summation in k, we obtain

N∑k,ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ)Jkℓ(t) = λi(t)N∑k=1

det((λi(t)I − J(t))k|k)

= λi(t)∏j( =i)

(λi(t) − λj(t)),

so that the denominator of the second term in the above SDE is reduced. Here,we use Lemma A.2 in the last equation. Therefore, the SDE is rewritten as

dλi(t) =N∑

k,ℓ=1


dMkℓ(t) −1

2λi(t)dt

+ τ∑j(=i)

1


which proves the claim.

Remark 3.1.3. By (3.7) and Cauchy-Schwarz inequality, Oii(t) ≥ 1 for t ≥ 0.When τ = 1, (3.4) is truly Dyson’s model (2.1) with β = 2. Indeed, theoverlaps (3.7) are sensitive for the normality of matrices as shown below.

Proposition 3.1.4. For τ = 1, Oij(t) ≡ δij.

Proof. J(t) is Hermitian with τ = 1, and so each of the matrices

(λi(t)I − J(t))(λj(t)I − J(t))∗ = (λi(t)I − J(t))(λj(t)I − J(t))

has the real eigenvalues (λi(t) − λk(t))(λj(t) − λk(t)), k = 1, . . . , N . If i = j,the matrix has only one zero eigenvalue, and for the numerator of (3.7), byLemma A.2 we obtain

N∑k=1

det((

(λi(t)I − J(t))(λj(t)I − J(t))∗)k|k

)=∏k(=i)

(λi(t) − λk(t))2.


On the other hand, for i = j, the matrix has two zero eigenvalues. Hence thissummation vanishes.

By Proposition 3.1.4 and (3.5)-(3.7), we have ⟨λRi , λRj ⟩t = tδij and ⟨λIi ⟩t ≡ 0for τ = 1. Hence the martingale terms of λ1(t), . . . , λN(t) are independentBrownian motions [37], and we get Dyson’s model. Similarly, we also obtainDyson’s model on the imaginary axis with τ = −1.

The parameter τ implies the Hermiticity and controls the speeds of realand imaginary parts of λi(t). Theorem 3.1.1 shows that for τ = 0, the driftterms of λi(t) completely vanish, and this fact is observed in the previous studyof GE. Moreover, (3.5) states that each trajectory of the eigenvalue processesof GE is Brownian motion, whereas they never collide.

Corollary 3.1.5. Only for τ = 0, each of the eigenvalue processes is conformalmartingale. Hence for each i = 1, . . . , N , let

Ti(t) := infu ≥ 0;Oii(u) > 2t,

then λi(Ti(t)) is a complex Brownian motion on (Ω,F , FTi(t)t≥0,P).

In the case of the matrix size N = 2, the numerical experiment of therelation between the distance of the two eigenvalues |λ1(t) − λ2(t)| and thediagonal overlap O11(t) was observed in [7]. They reported that O11(t), whichis the speed of the eigenvalue processes, takes a larger value as the two eigen-value processes get closer to each other. We attempt to justify this observationas follows. By (3.7) we have the explicit forms of the overlaps in N = 2:

O11(t) =||J(t)||22 − λ1(t)λ2(t) − λ1(t)λ2(t)

|λ1(t) − λ2(t)|2,

O12(t) =|λ1(t)|2 + |λ2(t)|2 − ||J(t)||22

|λ1(t) − λ2(t)|2,

(3.8)

where ||J(t)||22 :=∑2

i,j=1 |Jij(t)|2. The following relations hold:

O11(t) = O22(t), O12(t) = O21(t), O11(t) + O12(t) = 1. (3.9)

We note that O11(t) and O12(t) are real valued process for N = 2, neverthelessOij(t) are complex valued process forN ≥ 3. By (3.9), we need only to considerthe diagonal overlap O11(t).

Corollary 3.1.6. For N = 2 and −1 < τ < 1, O11(t) satisfies the stochasticdifferential equation:

dO11(t) =dM11(t) +(2O11(t) − 1)2 + 1

|λ1(t) − λ2(t)|2dt

3.2. PROOFS OF MAIN RESULTS 37

− τ(2O11(t) − 1)

(1

(λ1(t) − λ2(t))2+

1

(λ1(t) − λ2(t))2

)dt, t ≥ 0,

where M11 is local martingale and

d⟨O11⟩t =4O11(t)(2O11(t) − 1)(O11(t) − 1)

|λ1(t) − λ2(t)|2dt

− 2τO11(t)(O11(t) − 1)

(1

(λ1(t) − λ2(t))2+

1

(λ1(t) − λ2(t))2

)dt.

Moreover, the quadratic variation of O11(t) and |λ1(t) − λ2(t)|2 is

⟨O11, |λ1 − λ2|2⟩t = −8

∫ t

0

O11(s)(O11(s) − 1)ds ≤ 0.

Remark 3.1.7. For τ = 1 and τ = −1, J(t) is normal and O11(t) ≡ 1, sothat the negative correlation between O11(t) and |λ1(t) − λ2(t)|2 vanishes. Inparticular, for the deterministic parameter τ , O11(t) takes the maximum valueat τ = 0. To show this, we simply deal with the case of the initial conditionthat each of the eigenvalues starts at the origin, or equivalently, static EGEwhose entries are given by independent centered Gaussians with variance t.By Schur decomposition, there exists a unitary matrix U(t) such that

U(t)∗J(t)U(t) =

(λ1(t)

√1 − τ 2X

0 λ2(t)

),

where X is a complex Gaussian with mean 0 and variance t. Using this, we

have ||J(t)||22 = tr(J(t)J(t)∗

) (d)= |λ1(t)|2 + |λ2(t)|2 + (1 − τ 2)|X|2 and

O11(t) =||J(t)||22 − 2Re(λ1(t)λ2(t))

|λ1(t) − λ2(t)|2(d)= 1 +

t(1 − τ 2)Y

|λ1(t) − λ2(t)|2,

where Y is independent of λ1(t), λ2(t) and obeys exponential distribution withparameter 1. We notice that a similar deformation is obtained in [10] forGE. Consequently, the complete non-normality at τ = 0 provides the biggestnegative correlation of O11(t) and |λ1(t) − λ2(t)|2 and effects the behavior ofthe two eigenvalues significantly.

3.2 Proofs of main results

3.2.1 Proof of Theorem 3.1.1

Firstly, we derive the stochastic differential equations (3.4) by implicit functiontheorem until the first collision time t ∈ [0, Tcol), and finally we show Tcol =


∞, a.s. The detail calculations are summarized in the proof of Lemma 3.2.1.We define the N ×N deterministic matrix J as

J =

√1 + τ√

2H1 +

√−1

√1 − τ√

2H2, −1 ≤ τ ≤ 1,

where H1, H2 are Hermitian:

(H1)kℓ :=

xkk k = ℓxkℓ+

√−1ykℓ√2

k < ℓ

(H1)ℓk k > ℓ

, (H2)kℓ :=

αkk k = ℓαkℓ+

√−1βkℓ√2

k < ℓ

(H2)ℓk k > ℓ

.

Here, xkℓ, αkℓ (1 ≤ k ≤ ℓ ≤ N), ykℓ, βkℓ (1 ≤ k < ℓ ≤ N) are real deterministicvariables. Let f : R2N2+2 → C ∼= R2 be the characteristic polynomial of J :

f(J, λ) = f(λ) = f(λR, λI) := det(λI − J),

where λ := λR +√−1λI ∈ C. We denote f = fR +

√−1f I and the partial

derivative of f with respect to η by fη and also define fR := ∂f∂λR and fI := ∂f

∂λI .Assume that J has simple spectrum with the eigenvalues λ1, . . . , λN . f isanalytic with respect to λ, and so for all i = 1, . . . , N , the Jacobian is non-zero:

det

(∂f

∂(λR, λI)(λi)

)= det

(fRR (λi) fR

I (λi)f IR(λi) f I

I (λi)

)= |fλ(λi)|2 =

∏j(=i)

|λi − λj|2 = 0.

Hence we can apply implicit function theorem for each λi, and we obtain thederivatives of λi by using those of f as follows:

∂λRi∂η

= −Re(fη(λi)fλ(λi)

)|fλ(λi)|2

,∂λIi∂η

= −Im(fη(λi)fλ(λi)

)|fλ(λi)|2

. (3.10)

By using (3.10), we can also calculate the second derivatives of λRi and λIiwhich give us the drift terms in the stochastic differential equations (3.4) andthe quadratic variations in (3.5) and (3.6). For a C2 function g : RN2 → R,we denote the gradient and Laplacian of g by

∇g :=

(∂g

∂x11,∂g

∂x12, . . . ,

∂g

∂xNN

,∂g

∂α11

,∂g

∂α12

, . . . ,∂g

∂αNN

,∂g

∂y12, . . . ,

∂g

∂yN−1N

,

∂g

∂β12, . . . ,

∂g

∂βN−1N

),

∆g :=N∑k=1

( ∂2

∂x2kk+

∂2

∂α2kk

)g +

∑k<ℓ

( ∂2

∂x2kℓ+

∂2

∂y2kℓ+

∂2

∂α2kℓ

+∂2

∂β2kℓ

)g.

In the notation, a key lemma holds as following.


Lemma 3.2.1. For i = 1, . . . , N , λRi and λIi are C2 functions. We have

∆λRi = τRe(fλ(λi)fλλ(λi))

|fλ(λi)|2, ∆λIi = τ

Im(fλ(λi)fλλ(λi))

|fλ(λi)|2. (3.11)

Moreover, the inner products of the gradients of λRi and λIi are following:

∇λRi · ∇λRi =1

2

(∑Nk,ℓ=1

∣∣∣det((λiI − J)k|ℓ

)∣∣∣2|fλ(λi)|2

+ τ

),

∇λIi · ∇λIi =1

2

(∑Nk,ℓ=1


)∣∣∣2|fλ(λi)|2

− τ

),

∇λRi · ∇λIi = 0,

(3.12)

∇λRi · ∇λRj = ∇λIi · ∇λIj

=1

2Re

∑Nk=1 det

(((λiI − J)(λjI − J)∗)k|k

)fλ(λi)fλ(λj)

,

−∇λRi · ∇λIj = ∇λIi · ∇λRj

=1

2Im

∑Nk=1 det


)fλ(λi)fλ(λj)

, i = j.

(3.13)

The proof of Lemma 3.2.1 is given in Subsection 3.2.2. As a result, we canderive the stochastic differential equations (3.4) and the quadratic variations(3.5)-(3.7). Under the assumption that J has simple spectrum, we are ableto use the above calculus and apply Ito’s formula for λRi and λIi until firstcollision time Tcol defined as (2.2). Up to the time Tcol, we have

dλi(t) = dλRi (t) +√−1dλIi (t)

=N∑k=1

(∂λRi∂xkk

+√−1

∂λIi∂xkk

)dBkk(t) +

N∑k=1

(∂λRi∂αkk

+√−1

∂λIi∂αkk

)dbkk(t)

+∑k<ℓ

(∂λRi∂xkℓ

+√−1

∂λIi∂xkℓ

)dBR

kℓ(t) +

(∂λRi∂ykℓ

+√−1

∂λIi∂ykℓ

)dBI

kℓ(t)

+∑k<ℓ

(∂λRi∂αkℓ

+√−1

∂λIi∂αkℓ

)dbRkℓ(t) +

(∂λRi∂βkℓ

+√−1

∂λIi∂βkℓ

)dbIkℓ(t)

+

1

2

(∆λRi (t) +

√−1∆λIi (t)

)dt. (3.14)


For the local martingale part of (3.14), we have∂λR

i

∂η+

√−1

∂λIi

∂η= − fη(λi)

fλ(λi)by

(3.10). Using (3.26), (3.27) and (3.29) in Subsection 3.2.2, we find(∂λRi∂xkk

+√−1

∂λIi∂xkk

)dBkk(t) +

(∂λRi∂αkk

+√−1

∂λIi∂αkk

)dbkk(t)

=det((λi(t)I − J(t))k|k

)fλ(λi(t))

dJkk(t),(∂λRi∂xkℓ

+√−1

∂λIi∂xkℓ

)dBR

kℓ(t) +

(∂λRi∂ykℓ

+√−1

∂λIi∂ykℓ

)dBI

kℓ(t)

+

(∂λRi∂αkℓ

+√−1

∂λIi∂αkℓ

)dbRkℓ(t) +

(∂λRi∂βkℓ

+√−1

∂λIi∂βkℓ

)dbIkℓ(t)

=1

fλ(λi(t))

((−1)k+ℓ det

((λi(t)I − J(t))k|ℓ

)Jkℓ(t)

+ (−1)k+ℓ det((λi(t)I − J(t))ℓ|k

)Jℓk(t)

).

Therefore, the local martingale part of (3.14) is

N∑k,ℓ=1

(−1)k+ℓ det((λi(t)I − J(t))k|ℓ

)∏j(=i)(λi(t) − λj(t))

dJkℓ(t). (3.15)

For the drift part of (3.14), by (3.11) and Lemma A.1, we have

1

2

(∆λRi (t) +

√−1∆λIi (t)

)= τ

1

2

fλλ(λi(t))

fλ(λi(t))= τ

∑j(=i)

1

λi(t) − λj(t). (3.16)

From (3.14)-(3.16), we obtain the stochastic differential equations (3.4) fort ∈ [0, Tcol). Next, we derive the quadratic variations (3.5)-(3.7). Becausethe local martingale part of λRi and λIi are constructed by 2N2 independentBrownian motions as (3.14), we immediately find that

d⟨λRi ⟩t = ∇λRi (t) · ∇λRi (t)dt, d⟨λIi ⟩t = ∇λIi (t) · ∇λIi (t)dt,d⟨λi, λ

♯j⟩t = ∇λi(t) · ∇λ

♯j(t)dt, , ♯ = R, I.

By Lemma A.4, we rewrite the summation of the numerator of (3.12) as

N∑k,ℓ=1


)∣∣∣2 =N∑

k,ℓ=1

det((λiI − J)k|ℓ

)det((λiI − J)∗ℓ|k

)=

N∑k=1

det(

((λiI − J)(λiI − J)∗)k|k

).

Therefore, by (3.12) and (3.13), (3.5)-(3.7) hold. At the end of the proof ofTheorem 3.1.1, we show that the stochastic differential equations (3.4) actuallyhold for t ∈ [0,∞).


Proposition 3.2.2. Under the condition (3.3), Tcol = ∞, a.s.

Proof. Rogers and Shi showed that collision times are infinity almost surely forgeneral stochastic differential equations which include Dyson’s model in [57].Hence we have only to show the claim for −1 < τ < 1. However, their methoddoes not work for our complex eigenvalue processes straightforward becausethe martingale terms have the correlations (3.5)-(3.7). Accordingly, we referto the method in [11, 43]. Assume that Tcol < ∞ and define U : CN → C asfollows:

U(z1, . . . , zn) :=∏i<j

1

zi − zj. (3.17)

U is an analytic function with respect to zi, i = 1, . . . , N , and the derivativesof U are

∂ziU = −∑j(=i)

1

zi − zjU,

∂zi∂ziU = 2∑j( =i)

1

(zi − zj)2U + 2

∑j<kj,k =i

1

(zi − zj)(zi − zk)U.

From (3.5) and (3.6), we obtain the complex quadratic variations of eigenvalueprocesses as

d⟨λi, λj⟩t = τδijdt, d⟨λi, λj⟩t = Oij(t)dt. (3.18)

We denote dλ(i)t = dM

(i)t + τ

∑j(=i)

1

λ(i)t −λ

(j)t

dt, i = 1, . . . , N . Applying Lemma

A.5 to U and the eigenvalue processes for t ∈ [0, Tcol) with (3.18), we obtain

dU(λλλt) =N∑i=1

∂ziU(λλλt)dλ(i)t +

τ

2

N∑i=1

∂zi∂ziU(λλλt)dt

= −U(λλλt)N∑i=1

∑j(=i)

1

λ(i)t − λ

(j)t

dM(i)t − τU(λλλt)

N∑i=1

∑j(=i)

1

λ(i)t − λ

(j)t

2

dt

+τ

2U(λλλt)

N∑i=1

2∑j(=i)

1

(λ(i)t − λ

(j)t )2

+ 2∑j<kj,k =i

1

(λ(i)t − λ

(j)t )(λ

(i)t − λ

(k)t )

dt

= −U(λλλt)N∑i=1

∑j(=i)

1

λ(i)t − λ

(j)t

dM(i)t − τU(λλλt)

N∑i=1

∑j<kj,k =i

1

(λ(i)t − λ

(j)t )(λ

(i)t − λ

(k)t )

dt.

Indeed, the last summation vanishes.


Lemma 3.2.3.

N∑i=1

∑j<kj,k =i

1

(zi − zj)(zi − zk)= 0.

Proof.

(LHS) =∑i<j<k

1

(zi − zj)(zi − zk)+∑j<i<k

1

(zi − zj)(zi − zk)+∑j<k<i

1

(zi − zj)(zi − zk)

=∑i<j<k

(1

(zi − zj)(zi − zk)− 1

(zi − zj)(zj − zk)+

1

(zk − zi)(zk − zj)

)= 0.

Therefore, we find dU(λλλt) = −U(λλλt)∑N

i=1

∑j(=i)

1

λ(i)t −λ

(j)t

dM(i)t , t ∈ [0, Tcol),

and so U(λλλt) is a complex local martingale. We rewrite U(λλλt) = URt +

√−1U I

t .In the limit of t→ Tcol, by definition of U the radial part diverges. Hence eitherone of the divergence holds:

• limt→Tcol|UR

t | = ∞,

• limt→Tcol|U I

t | = ∞.

In the former case, URt is a one-dimensional local martingale whose real quadratic

variation is

⟨UR⟩t =1

2

∫ t

0

(d⟨U,U⟩s +

d⟨U,U⟩s + d⟨U,U⟩s2

)ds

=1

2

∫ t

0

(|U(λλλs)|2

∑i

∣∣∣∑k(=i)

1

λ(i)s − λ

(k)s

∣∣∣2Oii(s) + 2τRe

(U(λλλs)

2∑k( =i)

1

(λ(i)s − λ

(k)s )2

)

+ |U(λλλs)|2∑i =j

(∑k(=i)

1

λ(i)s − λ

(k)s

)(∑ℓ( =j)

1

λ(j)

s − λ(ℓ)

s

)Oij(s)

)ds.

We define TR(t) := infu ≥ 0; ⟨UR⟩u > t, and so the time changed processBR

t := URTR(t), t ∈ [0, ⟨UR⟩Tcol

) is a one-dimensional Brownian motion in theusual manner. Hence

limt→⟨UR⟩Tcol

|BRt | = lim

t→Tcol

|URt | = ∞

which never occur by the properties of Brownian motion’s paths [37]. In thelatter case, we also have the same contradiction by applying time change toU It with the real quadratic variation

⟨U I⟩t


=1

2

∫ t

0

(|U(λλλs)|2

∑i

∣∣∣∑k(=i)

1

λ(i)s − λ

(k)s

∣∣∣2Oii(s) − 2τRe

(U(λλλs)

2∑k(=i)

1

(λ(i)s − λ

(k)s )2

)

+ |U(λλλs)|2∑i =j

(∑k(=i)

1

λ(i)s − λ

(k)s

)(∑ℓ( =j)

1

λ(j)

s − λ(ℓ)

s

)Oij(s)

)ds.

Therefore, we have the contradiction in both cases, and the claim holds.

Together with Proposition 3.2.2, we complete the proof of Theorem 3.1.1.

3.2.2 Proof of Lemma 3.2.1

To show Lemma 3.2.1, we need to calculate the first and second derivatives ofλRi and λIi which are described by those of f as a result of implicit functiontheorem. For η = xkℓ, αkℓ, ykℓ, βkℓ, we apply chain rule to the first derivativesof λRi and λIi in (3.10) and obtain

∂2λRi∂η2

= −Re(fηη(λi)fλ(λi)

)+ 2(λRi,η)

2Re(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2λRi,ηλ

Ii,ηIm

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2Re

(fλ(λi)

2fλη(λi)fη(λi)

)+ Re

(fλ(λi)fλλ(λi)

)|fη(λi)|2

|fλ(λi)|4,

(3.19)

∂2λIi∂η2

= −Im(fηη(λi)fλ(λi)

)− 2(λIi,η)

2Im(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2λRi,ηλ

Ii,ηRe

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2Im

(fλ(λi)

2fλη(λi)fη(λi)

)− Im

(fλ(λi)fλλ(λi)

)|fη(λi)|2

|fλ(λi)|4.

(3.20)

Taking the summation in (3.19) and (3.20) for η = xkℓ, αkℓ, ykℓ, βkℓ, we yield

∆λRi = −Re(∆f(λi)fλ(λi)

)+ 2∇λRi · ∇λRi Re

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2∇λRi · ∇λIi Im

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2Re

(fλ(λi)

2∇fλ(λi) · ∇f(λi)

)|fλ(λi)|4

+Re(fλ(λi)fλλ(λi)

) (∇fR(λi) · ∇fR(λi) + ∇f I(λi) · ∇f I(λi)

)|fλ(λi)|4

,

(3.21)


∆λIi = −Im(∆f(λi)fλ(λi)

)− 2∇λIi · ∇λIi Im

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2∇λRi · ∇λIi Re

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+2Im

(fλ(λi)

2∇fλ(λi) · ∇f(λi)

)|fλ(λi)|4

−Im(fλ(λi)fλλ(λi)


)|fλ(λi)|4

.

(3.22)

From (3.10), we also have the gradient terms of λRi and λIi :

∇λRi · ∇λRi =fRR (λi)

2∇fR(λi) · ∇fR(λi)

|fλ(λi)|4

+Im(fλ(λi)

2)∇fR(λi) · ∇f I(λi) + f I

R(λi)2∇f I(λi) · ∇f I(λi)

|fλ(λi)|4(3.23)

∇λIi · ∇λIi =f IR(λi)

2∇fR(λi) · ∇fR(λi)

|fλ(λi)|4

+−Im

(fλ(λi)

2)∇fR(λi) · ∇f I(λi) + fR

R (λi)2f I(λi) · ∇f I(λi)

|fλ(λi)|4(3.24)

∇λRi · ∇λIi =Im(fλ(λi)

2∇f(λi) · ∇f(λi)

)2|fλ(λi)|4

. (3.25)

To calculate the above these quantities, we must know the derivatives of fexplicitly, and so we use Lemma A.3 with A = λI − J . We note that fork < ℓ, each determinant in (A.3) does not have (k, k) and (k, ℓ) entries, andwe obtain

fxkk= −

√1 + τ√

2det((λI − J)k|k

), fxkk,xkk

= 0. (3.26)

For the off-diagonal entries of A,

akℓaℓk =(1 + τ)(x2kℓ + y2kℓ) − (1 − τ)(α2

kℓ + β2kℓ) +

√−1 2

√1 − τ 2(xkℓαkℓ + ykℓβkℓ)

4

which gives

fxkℓ= −(1 + τ)2xkℓ +

√−1 2

√1 − τ 2αkℓ

4det(Akℓ|ℓk)

−∑q =k,ℓq<ℓ

(−1)k+q−1

√1 + τ

2aℓq det(Akℓ|ℓq) −

√1 + τ

2

∑q =k,ℓq>ℓ

(−1)k+qaℓq det(Akℓ|ℓq)


−√

1 + τ

2

∑p =k,ℓp>k

(−1)ℓ+p−1akp det(Akℓ|pk) −√

1 + τ

2

∑p =k,ℓp<k

(−1)ℓ+pakp det(Akℓ|pk)

= −√

1 + τ

2(−1)k+ℓ

(det((λI − J)k|ℓ

)+ det

((λI − J)ℓ|k

)). (3.27)

Expanding det((λI−J)k|ℓ

)and det

((λI−J)ℓ|k

)by each the ℓ-th and the k-th

row, we have

∂ det((λI − J)k|ℓ

)∂xkℓ

=

√1 + τ

2(−1)k+ℓ det

((λI − J)kℓ|kℓ

),

∂ det((λI − J)ℓ|k

)∂xkℓ

=

√1 + τ

2(−1)k+ℓ det

((λI − J)kℓ|kℓ

),

and we yield

fxkℓ,xkℓ= −1 + τ

2det((λI − J)kℓ|kℓ

). (3.28)

Similarly, we also have the other first and second derivatives of f :

fykℓ = −√−1

√1 + τ

2(−1)k+ℓ


)− det

((λI − J)ℓ|k

)),

fykℓ,ykℓ = fxkℓ,xkℓ,

fαkk= −

√−1

√1 − τ√

2det((λI − J)k|k

), fαkk,αkk

= 0,

fαkℓ= −

√−1

√1 − τ

2(−1)k+ℓ


)+ det

((λI − J)ℓ|k

)),

fαkℓ,αkℓ=

1 − τ

2det((λI − J)kℓ|kℓ

),

fβkℓ=

√1 − τ

2(−1)k+ℓ


)− det

((λI − J)ℓ|k

)), fβkℓ,βkℓ

= fαkℓ,αkℓ.

(3.29)

Using these derivatives and applying Lemma A.4, we get

∇f(λ) · ∇f(λ) = τ

N∑k,ℓ=1

det((λI − J)k|ℓ

)det((λI − J)ℓ|k

)= τ

N∑k=1

det((λI − J)2k|k

), (3.30)

∇f(λ) · ∇f(λ) =N∑

k,ℓ=1

∣∣∣det((λI − J)k|ℓ

)∣∣∣2. (3.31)

Indeed, the summation of (3.30) has a useful expression.


Lemma 3.2.4. For all λ ∈ C,

N∑k=1

det((λI − J)2k|k

)= fλ(λ)2 − 2

∑k<l

(λ− λk)(λ− λl)∏

m(=k,l)

(λ− λm)2.

In particular, if λi is one of the eigenvalues of J , then

N∑k=1

det((λiI − J)2k|k

)= fλ(λi)

2. (3.32)

Proof. (λI − J)2 has the eigenvalues (λ − λk)2, k = 1, . . . , N and by LemmaA.2, we obtain

N∑k=1

det((λI − J)2k|k

)=

N∑k=1

∏ℓ(=k)

(λ− λℓ)2. (3.33)

On the other hand, since fλ(λ) =∑N

k=1

∏ℓ(=k)(λ− λℓ), the claim holds.

Differentiating (3.30) with respect to λ and using (3.33), we have

∇fλ(λ) · ∇f(λ) = τ∑k =ℓ

(λ− λℓ)∏

m(=k,ℓ)

(λ− λm)2.

Because fλλ(λi) = 2∑

k(=i)

∏l(=i,k)(λi − λl), we take λ = λi and obtain

∇fλ(λi) · ∇f(λi) = τ∑k(=i)

(λi − λk)∏

l( =i,k)

(λi − λl)2 =

τ

2fλ(λi)fλλ(λi). (3.34)

Next, we calculate the Laplacian of f . Using the second derivatives in (3.26),(3.28) and (3.29),

∆f(λ) = −∑k<ℓ

(1 + τ) det((λI − J)kℓ|kℓ

)+∑k<ℓ

(1 − τ) det((λI − J)kℓ|kℓ

)= −2τ

∑k<ℓ

det((λI − J)kℓ|kℓ

).

We apply Lemma A.2 and obtain∑

k<ℓ det((λI−J)kℓ|kℓ

)=∑

k =ℓ

∏m( =k,ℓ)(λ−

λm). Taking λ = λi, we find the simple form:

∆f(λi) = −2τ∑k(=i)

∏l(=i,k)

(λi − λl) = −τfλλ(λi). (3.35)

Using all the above equations, we prove Lemma 3.2.1.


Proof of Lemma 3.2.1. Firstly, we show (3.11). From (3.30) and (3.32), thenumerator of (3.25) vanishes, and we get

∇λRi · ∇λIi = 0. (3.36)

For (3.21), by (3.34), (3.35) and (3.36), we have

∆λRi = −−τRe

(fλ(λi)fλλ(λi)

)+ 2∇λRi · ∇λRi Re

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+τRe

(fλ(λi)

2fλ(λi)fλλ(λi)

)|fλ(λi)|4

+Re(fλ(λi)fλλ(λi)


)|fλ(λi)|4

=Re(fλ(λi)fλλ(λi)

)|fλ(λi)|2

×(

2τ − 2∇λRi · ∇λRi +∇fR(λi) · ∇fR(λi) + ∇f I(λi) · ∇f I(λi)

|fλ(λi)|2

).

To calculate the last term, we rewrite (3.30) by using fR(λ) and f I(λ) as

∇f(λ) · ∇f(λ) = ∇fR(λ) · ∇fR(λ) −∇f I(λ) · ∇f I(λ) + 2√−1∇fR(λ) · ∇f I(λ).

Applying (3.30) and (3.32), we have

∇fR(λi) · ∇fR(λi) −∇f I(λi) · ∇f I(λi) = τRe(fλ(λi)2),

∇fR(λi) · ∇f I(λi) =τ

2Im(fλ(λi)

2)(3.37)

which gives

− 2∇λRi · ∇λRi +∇fR(λi) · ∇fR(λi) + ∇f I(λi) · ∇f I(λi)

|fλ(λi)|2= −τ. (3.38)

Here, we use (3.23). Hence we obtain the former equation of (3.11). Similarly,we calculate (3.22) and get

∆λIi = −−τ Im

(fλ(λi)fλλ(λi)

)− 2∇λIi · ∇λIi Im

(fλ(λi)fλλ(λi)

)|fλ(λi)|2

+τ Im

(fλ(λi)

2fλ(λi)fλλ(λi)

)|fλ(λi)|4

−Im(fλ(λi)fλλ(λi)


)|fλ(λi)|4


=Im(fλ(λi)fλλ(λi)

)|fλ(λi)|2

×(

2τ + 2∇λIi · ∇λIi −∇fR(λi) · ∇fR(λi) + ∇f I(λi) · ∇f I(λi)

|fλ(λi)|2

).

From (3.24) and (3.37), we also yield

2∇λIi · ∇λIi −∇fR(λi) · ∇fR(λi) + ∇f I(λi) · ∇f I(λi)

|fλ(λi)|2= −τ. (3.39)

Hence we obtain the latter equation of (3.11), and we finish the calculationsof ∆λRi and ∆λIi . Secondly, we show (3.12). Note that

∇f(λ) · ∇f(λ) = ∇fR(λ) · ∇fR(λ) + ∇f I(λ) · ∇f I(λ),

and by (3.31), (3.38), (3.39) and (3.36), we yield (3.12). Finally, we show(3.13). From (3.10), for i = j, we have

∇λRi · ∇λRj =fRR (λi)f

RR (λj)∇fR(λi) · ∇fR(λj) + fR

R (λi)fIR(λj)∇fR(λi) · ∇f I(λj)

|fλ(λi)|2|fλ(λj)|2

+f IR(λi)f

RR (λj)∇f I(λi) · ∇fR(λj) + f I

R(λi)fIR(λj)∇f I(λi) · ∇f I(λj)

|fλ(λi)|2|fλ(λj)|2,

and so we want the gradient terms. By straight computation, we get

∇fR(λi) · ∇fR(λj) =1

2Re

(N∑k=1

det(

((λiI − J)(λjI − J)∗)k|k

))

+τ

2Re

(N∑k=1

det(

((λiI − J)(λjI − J))k|k

)).

Here, we use det((λjI − J)i|j

)= det

((λjI − J)∗j|i

)and Lemma A.4. Since

(λiI − J)(λjI − J) must have two zero eigenvalues, the second summationvanishes by applying Lemma A.2. Therefore, we obtain

∇fR(λi) · ∇fR(λj) =1

2Re

(N∑k=1

det(


)). (3.40)

Similarly, we also obtain

∇f I(λi) · ∇f I(λj) = ∇fR(λi) · ∇fR(λj), (3.41)

−∇fR(λi) · ∇f I(λj) = ∇f I(λi) · ∇fR(λj)


=1

2Im

(N∑k=1

det(


)). (3.42)

By (3.40)-(3.42), we conclude

∇λRi · ∇λRj =1

2Re

∑Nk=1 det


)fλ(λi)fλ(λj)

.

The others are also obtained by the same calculation:

∇λIi · ∇λIj = ∇λRi · ∇λRj ,

−∇λRi · ∇λIj = ∇λIi · ∇λRj =1

2Im

∑Nk=1 det


)fλ(λi)fλ(λj)

.

Hence we also obtain (3.13), and we finish the proof of Lemma 3.2.1.

3.2.3 Proof of Corollary 3.1.6

From (3.8), (3.9) and (3.18), we apply Ito’s formula to O11 and obtain

d(||J ||22 − λ1λ2 − λ1λ2

)(t) =

2∑i,j=1

(Jij(t)dJij(t) + Jij(t)dJij(t)) + 2(O11(t) + 1)dt

− λ1(t)dλ2(t) − λ2(t)dλ1(t) − λ1(t)dλ2(t) − λ2(t)dλ1(t),

d|λ1 − λ2|2(t) = (λ1(t) − λ2(t))d(λ1(t) − λ2(t)) + (λ1(t) − λ2(t))d(λ1(t) − λ2(t))

+2(2O11(t) − 1)dt,

dO11(t) =d(||J ||22 − λ1λ2 − λ1λ2

)(t) −O11(t)d|λ1 − λ2|2(t)

|λ1(t) − λ2(t)|2

+O11(t)d⟨|λ1 − λ2|2⟩t − d⟨||J ||22 − λ1λ2 − λ1λ2, |λ1 − λ2|2⟩t

|λ1(t) − λ2(t)|4.

(3.43)

We first deal with the local martingale part of (3.43). From the above twostochastic differential equations and (3.4), we have

(local martingale term of dO11)

=2

|λ1(t) − λ2(t)|2Re

( 2∑i,j=1

Jij(t)dJij(t)


−O11(t)λ1(t) − λ2(t)

λ1(t) − λ2(t)

(2J21(t)dJ12(t) + 2J12(t)dJ21(t)

)−O11(t)

λ1(t) − λ2(t)

λ1(t) − λ2(t)

((λ1(t) + λ2(t) − 2J22(t))dJ11(t)

+ (λ1(t) + λ2(t) − 2J11(t))dJ22(t))

+λ1(t)

λ1(t) − λ2(t)

((λ2(t) − J22(t))dJ11(t) + J21(t)dJ12(t)

+ J12(t)dJ21(t) + (λ2(t) − J11(t))dJ22(t))

− λ2(t)

λ1(t) − λ2(t)

((λ1(t) − J22(t))dJ11(t) + J21(t)dJ12(t)

+ J12(t)dJ21(t) + (λ1(t) − J11(t))dJ22(t)))

=2

|λ1(t) − λ2(t)|2Re

( 2∑i,j=1

Jij(t)dJij(t)

+ (2O11(t) − 1)λ1(t) − λ2(t)

λ1(t) − λ2(t)

(J22(t)dJ11(t) + J11(t)dJ22(t)

− J21(t)dJ12(t) − J12(t)dJ21(t))

+λ1(t)λ2(t) − λ1(t)λ2(t) −O11(t)(λ1(t) + λ2(t))(λ1(t) − λ2(t))

λ1(t) − λ2(t)(dJ11(t) + dJ22(t))

).

(3.44)

The above equation gives the explicit form of M11(t) in Corollary 3.1.6. Next,we calculate the drift terms of (3.43). From (3.9), (3.18), (3.2) and the ele-mentary relations

tr(J(t)) = λ1(t) + λ2(t), det(J(t)) = λ1(t)λ2(t),

we obtain the real quadratic variations:

d⟨|λ1 − λ2|2⟩t=(

4(2O11(t) − 1)|λ1(t) − λ2(t)|2 + 2τ((λ1(t) − λ2(t))

2 + (λ1(t) − λ2(t))2))dt,

(3.45)

d⟨||J ||22, λ1⟩t =1

λ1(t) − λ2(t)

×(

(λ1(t) − J22(t))J11(t) + J21(t)J12(t) + J12(t)J21(t) + (λ1(t) − J11(t))J22(t)

+ τ(λ1(t) − J22(t))J11(t) + τ |J21(t)|2 + τ |J12(t)|2 + τ(λ1(t) − J11(t))J22(t)

)dt


= λ1(t)dt+ τ||J(t)||22 − λ2(t)(λ1(t) + λ2(t))

λ1(t) − λ2(t)dt,

d⟨||J ||22, λ2⟩t = λ2(t)dt+ τ||J(t)||22 − λ1(t)(λ1(t) + λ2(t))

λ2(t) − λ1(t)dt.

By using the equation

2||J(t)||22 − |λ1(t) + λ2(t)|2 + |λ1(t) − λ2(t)|2 = 2O11(t)|λ1(t) − λ2(t)|2,

we get

d⟨||J ||22 − λ1λ2 − λ1λ2, |λ1 − λ2|2⟩t= d⟨||J ||22, |λ1 − λ2|2⟩t − d⟨λ1λ2, |λ1 − λ2|2⟩t − d⟨λ1λ2, |λ1 − λ2|2⟩t

= 2Re

(λ1(t) − λ2(t))

(λ1(t) − λ2(t) + τ

2||J(t)||22 − |λ1(t) + λ2(t)|2

λ1(t) − λ2(t)

)+ τ(λ1(t) − λ2(t))λ1(t) − τ(λ1(t) − λ2(t))λ2(t)

+ (2O11(t) − 1)(|λ1(t)|2 − 2λ1(t)λ2(t) + |λ2(t)|2)

dt

= 4O11(t)|λ1(t) − λ2(t)|2dt+ 2τO11(t)((λ1(t) − λ2(t))2 + (λ1(t) − λ2(t))

2)dt.(3.46)

From (3.4), (3.43), (3.45) and (3.46), we obtain

(drift term of dO11)

=1

|λ1(t) − λ2(t)|2

(2(O11(t) + 1) + 2τRe

(λ1(t) − λ2(t)

λ1(t) − λ2(t)

))dt

− O11(t)

|λ1(t) − λ2(t)|2

(2(2O11(t) − 1) + 4τRe

(λ1(t) − λ2(t)

λ1(t) − λ2(t)

))dt

+O11(t)

|λ1(t) − λ2(t)|4(

4(2O11(t) − 1)|λ1(t) − λ2(t)|2

+ 2τ((λ1(t) − λ2(t))2 + (λ1(t) − λ2(t))

2))dt

− 1

|λ1(t) − λ2(t)|4

(4O11(t)|λ1(t) − λ2(t)|2

+ 2τO11(t)((λ1(t) − λ2(t))2 + (λ1(t) − λ2(t))

2)

)dt

=1

|λ1(t) − λ2(t)|2

((2O11(t) − 1)2 + 1 − 2τ(2O11(t) − 1)Re

(λ1(t) − λ2(t)

λ1(t) − λ2(t)

))dt


which gives the drift terms of Corollary 3.1.6. Thirdly, we calculate the realquadratic variation ⟨O11⟩t. We denote

A := (2O11(t) − 1)λ1(t) − λ2(t)

λ1(t) − λ2(t),

B :=λ1(t)λ2(t) − λ1(t)λ2(t) −O11(t)(λ1(t) + λ2(t))(λ1(t) − λ2(t))

λ1(t) − λ2(t)

and rewrite (3.44) as

(local martingale term of dO11)

=1

|λ1(t) − λ2(t)|2

×(

(J11(t) + AJ22(t) +B)dJ11(t) + (J22(t) + AJ11(t) +B)dJ22(t)

+ (J12(t) − AJ21(t))dJ12(t) + (J21(t) − AJ12(t))dJ21(t)

+ (J11(t) + AJ22(t) +B)dJ11(t) + (J22(t) + AJ11(t) +B)dJ22(t)

+ (J12(t) − AJ21(t))dJ12(t) + (J21(t) − AJ12(t))dJ21(t)

).

By using (3.2) and the above notations,

d⟨O11⟩t =1

|λ1(t) − λ2(t)|4

×(τ(J11(t) + AJ22(t) +B)2 + τ(J22(t) + AJ11(t) +B)2 + τ(J11(t) + AJ22(t) +B)2

+ τ(J22(t) + AJ11(t) +B)2 + 2|J11(t) + AJ22(t) +B|2 + 2|J22(t) + AJ11(t) +B|2

+ 2|J12(t) − AJ21(t)|2 + 2τ(J12(t) − AJ21(t))(J21(t) − AJ12(t))

+ 2|J21(t) − AJ12(t)|2 + 2τ(J12(t) − AJ21(t))(J21(t) − AJ12(t))

)dt

=2

|λ1(t) − λ2(t)|4

(||J(t)||22(|A|2 + 1) + 2|B|2 + 2Aλ1(t)λ2(t) + 2Aλ1(t)λ2(t)

+B(λ1(t) + λ2(t) + A(λ1(t) + λ2(t))

)+B

(λ1(t) + λ2(t) + A(λ1(t) + λ2(t))

))dt

+2τ

|λ1(t) − λ2(t)|4

× Re

((A2 + 1)(J11(t)

2 + J22(t)2 + 2B

(λ1(t) + λ2(t) + A(λ1(t) + λ2(t))

)+ 2J12(t)J21(t)) + 2A(J11(t)J22(t) + J11(t)J22(t) − |J12(t)|2 − |J21(t)|2) + 2B2

)dt.


Because

− 2B = λ1(t) + λ2(t) + A(λ1(t) + λ2(t)),

J11(t)2 + J22(t)

2 + 2J12(t)J21(t) = λ1(t)2 + λ2(t)

2,

J11(t)J22(t) + J11(t)J22(t) − |J12(t)|2 − |J21(t)|2 = |λ1(t) + λ2(t)|2 − ||J(t)||22,

we obtain

first term of d⟨O11⟩t =4O11(t)(2O11(t) − 1)(O11(t) − 1)

|λ1(t) − λ2(t)|2dt,

second term of d⟨O11⟩t = −4τO11(t)(O11(t) − 1)Re

((λ1(t) − λ2(t))

2)

|λ1(t) − λ2(t)|4dt.

Combining both, we obtain d⟨O11⟩t. Finally, from (3.45) and (3.46), we yield

d⟨O11, |λ1 − λ2|2⟩t =d⟨||J ||22 − λ1λ2 − λ1λ2, |λ1 − λ2|2⟩t −O11(t)d⟨|λ1 − λ2|2⟩t

|λ1(t) − λ2(t)|2

=1

|λ1(t) − λ2(t)|2

×(

4O11(t)|λ1(t) − λ2(t)|2 + 2τO11(t)((λ1(t) − λ2(t))

2 + (λ1(t) − λ2(t))2)

−O11(t)(

4(2O11(t) − 1)|λ1(t) − λ2(t)|2 + 2τ((λ1(t) − λ2(t))

2 + (λ1(t) − λ2(t))2)))

dt

= −8O11(t)(O11(t) − 1)dt.

Therefore, we finish the proof of Corollary 3.1.6.

Appendix A

Tools and basic properties

In this appendix, we record some elementary properties regarding eigenvaluesand determinants. We also review complex Ito’s formula.

Lemma A.1 (Drift terms of Dyson’s model and characteristic polynomial).Assume that an N × N matrix A has simple spectrum λ1, . . . , λN , and thecharacteristic polynomial is f(λ) := det(λIN −A). Then for any i = 1, . . . , N ,

fλλ(λi)

fλ(λi)= 2

∑j(=i)

1

λi − λj. (A.1)

Proof. By definition of eigenvalues, f(λ) =∏N

j=1(λ−λj) = (λ−λi)∏

j(=i)(λ−λj). Differentiating this with respect to λ, we have

fλ(λ) =∏j(=i)

(λ− λj) + (λ− λi)∑j(=i)

∏k(=i,j)

(λ− λk),

fλλ(λ) =∑j(=i)

∏k(=i,j)

(λ− λk) +∑j(=i)

∏k(=i,j)

(λ− λk)

+ (λ− λi)∑j(=i)

∑k(=i,j)

∏ℓ(=i,j,k)

(λ− λℓ).

Substituting λ = λi,

fλ(λi) =∏j(=i)

(λi − λj), fλλ(λi) = 2∑j(=i)

∏k( =i,j)

(λi − λk).

On the other hand, the right hand side of (A.1) is∑j(=i)

1

λi − λj=

∑j(=i)

∏k(=i,j)(λi − λk)∏

j(=i)(λi − λj),

and the claim holds.

54

55

Lemma A.2 (Minor determinants and eigenvalues).Assume that an N × N matrix A has eigenvalues λ1, . . . , λN . Then for allk = 1, . . . , N , ∑

1≤j1<···<jk≤N

k∏ℓ=1

λjℓ =∑

1≤j1<···<jk≤N

det1≤ℓ,m≤k

(Ajℓjm), (A.2)

where det1≤ℓ,m≤k

(Ajℓjm) is the k-th principal minor indexed by j1 < · · · < jk ⊂1, . . . , N:

det1≤ℓ,m≤k

(Ajℓjm) = det

aj1j1 aj1j2 · · · aj1jkaj2j1 aj2j2 · · · aj2jk

.... . .

...ajkj1 ajkj2 · · · ajkjk

.

Proof. Applying binomial expansion to the characteristic polynomial f(λ) =det(λIN − A), we have

f(λ) = λN +N∑k=1

(−1)kλN−k∑

1≤j1<···<jk≤N

k∏ℓ=1

λjℓ .

We also apply Fredholm determinant expansion to f(λ) and obtain

f(λ) = λN +N∑k=1

(−1)kλN−k∑

1≤j1<···<jk≤N

det1≤ℓ,m≤k

(Ajℓjm).

Therefore, the claim holds by comparing the coefficient of λN−k on each other.Lemma A.3 (Twice cofactor expansion form).For an N ×N matrix A and fixed integers k < ℓ,

detA = akk det(Ak|k) − akℓaℓk det(Akℓ|ℓk) +∑q =k,ℓq<ℓ

(−1)k+q−1akℓaℓq det(Akℓ|ℓq)

+∑q =k,ℓq>ℓ

(−1)k+qakℓaℓq det(Akℓ|ℓq) +∑p =k,ℓp>k

(−1)ℓ+p−1akpaℓk det(Akℓ|pk)

+∑p=k,ℓp<k

(−1)ℓ+pakpaℓk det(Akℓ|pk) +∑

p =k,ℓ, q =kp>q

(−1)k+ℓ+p+q−1akpaℓq det(Akℓ|pq)

+∑

p=k,ℓ, q =kp<q

(−1)k+ℓ+p+qakpaℓq det(Akℓ|pq), (A.3)

where Akℓ|pq is the (N−2)×(N−2) minor matrix that is obtained by removingthe k, ℓ-th rows and the p, q-th columns from A.

56 APPENDIX A. TOOLS AND BASIC PROPERTIES

Proof. We expand detA by the k-th row, and we also expand each of the(N − 1)-th determinants by the ℓ-th row.

Lemma A.4 (Cauchy-Binet formula, [36]).Let A ∈ Mm,n(C). For index sets α = i1, . . . , ip ⊆ 1, . . . ,m, p ≤ m andβ = j1, . . . , jq ⊆ 1, . . . , n, q ≤ n, the p× q submatrix A(α, β) is defined asA(α, β)r,s := Air,js . Here, i1 < · · · < ip and j1 < · · · < jq hold and these indexsets are ordered lexicographically. When ♯α = ♯β = k ≤ minm,n, the k-thcompound matrix of A is defined as the

(mk

)×(nk

)matrix whose (α, β) entry

is det(A(α, β)) and we denote this by Ck(A).Let A ∈ Mm,k(C), B ∈ Mk,n(C) and C := AB. We fix the index sets α ⊆1, . . . ,m and β ⊆ 1, . . . , n where ♯α = ♯β = r ≤ minm, k, n. Then thedeterminant of the submatrix C(α, β) has an expression:

det(C(α, β)) =∑γ

det(A(α, γ)) det(B(γ, β)), (A.4)

where the summation is taken over all index sets γ ⊆ 1, . . . , k of cardinalityr.

Lemma A.5 (Ito’s formula for complex case).

Suppose that Zt = (Z(1)t , . . . , Z

(n)t ) is a continuous complex semi-martingale

vector and f : Cn → C is a C2 function. Then

df(Zt) =n∑

i=1

(∂zif(Zt)dZ

(i)t + ∂zif(Zt)dZ

(i)

t

)+

1

2

n∑i=1

(∂zi∂zif(Zt)d⟨Z(i), Z(i)⟩t + 2∂zi∂zif(Zt)d⟨Z(i), Z

(i)⟩t

+ ∂zi ∂zif(Zt)d⟨Z(i), Z

(i)⟩t)

+∑i<j

(∂zi∂zjf(Zt)d⟨Z(i), Z(j)⟩t + ∂zi∂zjf(Zt)d⟨Z

(i), Z(j)⟩t

+ ∂zi∂zjf(Zt)d⟨Z(i), Z(j)⟩t + ∂zi ∂zjf(Zt)d⟨Z

(i), Z

(j)⟩t), (A.5)

where the complex quadratic variation d⟨·, ·⟩t is defined by (2.8).

Proof. We apply standard Ito’s formula for f : R2n → R2 and put the real andimaginary part together algebraically.

Acknowledgments

The author would like to express his appreciation to his supervisor Prof.Hideki Tanemura for his valuable discussions, encouragement and insightfulcomments.

He would like to express his thanks to Prof. Takashi Imamura for hisgenerous supports and valuable comments.

He would like to express his thanks to Prof. Makoto Katori for his valuablecomments and encouragement. He would like to express his thanks to Dr.Mikio Shibuya and Dr. Syota Esaki for their continuous encouragement andvaluable discussions. He would like to express his gratitude to Prof. RyokiFukushima and Prof. Hirofumi Osada for their insightful comments. He wouldlike to express his gratitude to Prof. Noriyoshi Sakuma, Dr. Yoshihiro Abe, Dr.Yosuke Kawamoto and Dr. Kenkichi Tsunoda for their valuable discussionsand encouragement.

57

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